Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of "x" and "y" would be any expression of the form "ax" + "by", where "a" and "b" are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics.
Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Definition.
Let "V" be a vector space over the field "K". As usual, we call elements of "V" "vectors" and call elements of "K" "scalars".
If v1...,v"n" are vectors and "a"1...,"a""n" are scalars, then the "linear combination of those vectors with those scalars as coefficients" is
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of v1...,v"n" always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of linear dependence: a family "F" of vectors is linearly independent precisely if any linear combination of the vectors in "F" (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each v"i"; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.
In a given situation, "K" and "V" may be specified explicitly, or they may be obvious from context. In that case, we often speak of "a linear combination of the vectors" v1...,v"n", with the coefficients unspecified (except that they must belong to "K"). Or, if "S" is a subset of "V", we may speak of "a linear combination of vectors in S", where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set "S" (and the coefficients must belong to "K"). Finally, we may speak simply of "a linear combination", where nothing is specified (except that the vectors must belong to "V" and the coefficients must belong to "K"); in this case one is probably referring to the expression, since every vector in "V" is certainly the value of some linear combination.
Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalizations below).
However, the set "S" that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors.
Also, there is no reason that "n" cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in "V".
Examples and counterexamples.
Euclidean vectors.
Let the field "K" be the set R of real numbers, and let the vector space "V" be the Euclidean space R3.
Consider the vectors , and .
Then "any" vector in R3 is a linear combination of e1, e2, and e3.
To see that this is so, take an arbitrary vector ("a"1,"a"2,"a"3) in R3, and write:
Functions.
Let "K" be the set C of all complex numbers, and let "V" be the set CC("R") of all continuous functions from the real line R to the complex plane C.
Consider the vectors (functions) "f" and "g" defined by "f"("t") := "e""it" and "g"("t") := "e"−"it".
Some linear combinations of "f" and "g" are:
On the other hand, the constant function 3 is "not" a linear combination of "f" and "g". To see this, suppose that 3 could be written as a linear combination of "e""it" and "e"−"it". This means that there would exist complex scalars "a" and "b" such that for all real numbers "t". Setting "t" = 0 and "t" = π gives the equations and , and clearly this cannot happen. See Euler's identity.
Polynomials.
Let "K" be R, C, or any field, and let "V" be the set "P" of all polynomials with coefficients taken from the field "K".
Consider the vectors (polynomials) "p"1 := 1, , and .
Is the polynomial "x"2 − 1 a linear combination of "p"1, "p"2, and "p"3?
To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector "x"2 − 1.
Picking arbitrary coefficients "a"1, "a"2, and "a"3, we want
Multiplying the polynomials out, this means
and collecting like powers of "x", we get
Two polynomials are equal if and only if their corresponding coefficients are equal, so we can conclude
This system of linear equations can easily be solved.
First, the first equation simply says that "a"3 is 1.
Knowing that, we can solve the second equation for "a"2, which comes out to −1.
Finally, the last equation tells us that "a"1 is also −1.
Therefore, the only possible way to get a linear combination is with these coefficients.
Indeed,
so "x"2 − 1 "is" a linear combination of "p"1, "p"2, and "p"3.
On the other hand, what about the polynomial "x"3 − 1? If we try to make this vector a linear combination of "p"1, "p"2, and "p"3, then following the same process as before, we get the equation
However, when we set corresponding coefficients equal in this case, the equation for "x"3 is
which is always false.
Therefore, there is no way for this to work, and "x"3 − 1 is "not" a linear combination of "p"1, "p"2, and "p"3.
The linear span.
Take an arbitrary field "K", an arbitrary vector space "V", and let v1...,v"n" be vectors (in "V").
It’s interesting to consider the set of "all" linear combinations of these vectors.
This set is called the "linear span" (or just "span") of the vectors, say "S" = {v1, ..., v"n"}. We write the span of "S" as span("S") or sp("S"):
Linear independence.
For some sets of vectors v1...,v"n",
a single vector can be written in two different ways as a linear combination of them:
Equivalently, by subtracting these (formula_14) a non-trivial combination is zero:
If that is possible, then v1...,v"n" are called "linearly dependent"; otherwise, they are "linearly independent".
Similarly, we can speak of linear dependence or independence of an arbitrary set "S" of vectors.
If "S" is linearly independent and the span of "S" equals "V", then "S" is a basis for "V".
Affine, conical, and convex combinations.
By restricting the coefficients used in linear combinations, one can define the related concepts of affine combination, conical combination, and convex combination, and the associated notions of sets closed under these operations.
Because these are more "restricted" operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets are "generalizations" of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone.
These concepts often arise when one can take certain linear combinations of objects, but not any: for example, probability distributions are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and positive measures are closed under conical combination but not affine or linear – hence one defines signed measures as the linear closure.
Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over an ordered field (or ordered ring), generally the real numbers.
If one allows only scalar multiplication, not addition, one obtains a (not necessarily convex) cone; one often restricts the definition to only allowing multiplication by positive scalars.
All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.
Operad theory.
More abstractly, in the language of operad theory, one can consider vector spaces to be algebras over the operad formula_16 (the infinite direct sum, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector formula_17 for instance corresponds to the linear combination formula_18. Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by formula_19 being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.
From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that "all possible" algebraic operations in a vector space are linear combinations.
The basic operations of addition and scalar multiplication, together with the existence of an additive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are a generating set for the operad of all linear combinations.
Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces.
Generalizations.
If "V" is a topological vector space, then there may be a way to make sense of certain "infinite" linear combinations, using the topology of "V".
For example, we might be able to speak of "a"1v1 + "a"2v2 + "a"3v3 + ⋯, going on forever.
Such infinite linear combinations do not always make sense; we call them "convergent" when they do.
Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis.
The articles on the various flavors of topological vector spaces go into more detail about these.
If "K" is a commutative ring instead of a field, then everything that has been said above about linear combinations generalizes to this case without change.
The only difference is that we call spaces like this "V" modules instead of vector spaces.
If "K" is a noncommutative ring, then the concept still generalizes, with one caveat:
Since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module.
This is simply a matter of doing scalar multiplication on the correct side.
A more complicated twist comes when "V" is a bimodule over two rings, "K"L and "K"R.
In that case, the most general linear combination looks like
where "a"1...,"a""n" belong to "K"L, "b"1...,"b""n" belong to "K"R, and v1,…,v"n" belong to "V".
Application.
An important application of linear combinations is to wave functions in quantum mechanics.


Multiplicatively closed set

In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset "S" of a ring "R" such that the following two conditions hold:
In other words, "S" is closed under taking finite products, including the empty product 1.
Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring.
Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings.
A subset "S" of a ring "R" is called saturated if it is closed under taking divisors: i.e., whenever a product "xy" is in "S", the elements "x" and "y" are in "S" too.
Examples.
Common examples of multiplicative sets include:


Sieve (category theory)

In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introduced by in order to reformulate the notion of a Grothendieck topology.
Definition.
Let C be a category, and let "c" be an object of C. A sieve formula_1 on "c" is a subfunctor of Hom(−, "c"), i.e., for all objects "c"′ of C, "S"("c"′) ⊆ Hom("c"′, "c"), and for all arrows "f":"c"″→"c"′, "S"("f") is the restriction of Hom("f", "c"), the pullback by "f" (in the sense of precomposition, not of fiber products), to "S"("c"′); see the next section, below.
Put another way, a sieve is a collection "S" of arrows with a common codomain that satisfies the condition, "If "g":"c"′→"c" is an arrow in "S", and if "f":"c"″→"c"′ is any other arrow in C, then "gf" is in "S"." Consequently, sieves are similar to right ideals in ring theory or filters in order theory.
Pullback of sieves.
The most common operation on a sieve is "pullback". Pulling back a sieve "S" on "c" by an arrow "f":"c"′→"c" gives a new sieve "f"*"S" on "c"′. This new sieve consists of all the arrows in "S" that factor through "c"′.
There are several equivalent ways of defining "f"*"S". The simplest is:
A more abstract formulation is:
Here the map Hom(−, "c"′)→Hom(−, "c") is Hom("f", "c"′), the pullback by "f".
The latter formulation suggests that we can also take the image of "S"×Hom(−, "c")Hom(−, "c"′) under the natural map to Hom(−, "c"). This will be the image of "f"*"S" under composition with "f". For each object "d" of C, this sieve will consist of all arrows "fg", where "g":"d"→"c"′ is an arrow of "f"*"S"("d"). In other words, it consists of all arrows in "S" that can be factored through "f".
If we denote by ∅"c" the empty sieve on "c", that is, the sieve for which ∅("d") is always the empty set, then for any "f":"c"′→"c", "f"*∅"c" is ∅"c"′. Furthermore, "f"*Hom(−, "c") = Hom(−, "c"′).
Properties of sieves.
Let "S" and "S"′ be two sieves on "c". We say that "S" ⊆ "S"′ if for all objects "c"′ of C, "S"("c"′) ⊆ "S"′("c"′). For all objects "d" of C, we define ("S" ∪ "S"′)("d") to be "S"("d") ∪ "S"′("d") and ("S" ∩ "S"′)("d") to be "S"("d") ∩ "S"′("d"). We can clearly extend this definition to infinite unions and intersections as well.
If we define SieveC("c") (or Sieve("c") for short) to be the set of all sieves on "c", then Sieve("c") becomes partially ordered under ⊆. It is easy to see from the definition that the union or intersection of any family of sieves on "c" is a sieve on "c", so Sieve("c") is a complete lattice.
A Grothendieck topology is a collection of sieves subject to certain properties. These sieves are called "covering sieves". The set of all covering sieves on an object "c" is a subset "J"("c") of Sieve("c"). "J"("c") satisfies several properties in addition to those required by the definition:
Consequently, "J"("c") is also a distributive lattice, and it is cofinal in Sieve("c").


Profunctor

In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.
Definition.
A profunctor (also named by the French school and module by the Sydney school) formula_1 from a category formula_2 to a category formula_3, written
is defined to be a functor
where formula_6 denotes the opposite category of formula_3 and formula_8 denotes the category of sets. Given morphisms formula_9 respectively in formula_10 and an element formula_11, we write formula_12 to denote the actions.
Using the cartesian closure of formula_13, the category of small categories, the profunctor formula_14 can be seen as a functor
where formula_16 denotes the category formula_17 of presheaves over formula_3.
A correspondence from formula_19 to formula_20 is a profunctor formula_21.
Profunctors as categories.
An equivalent definition of a profunctor formula_4 is a category whose objects are the disjoint union of the objects of formula_2 and the objects of formula_3, and whose morphisms are the morphisms of formula_2 and the morphisms of formula_3, plus zero or more additional morphisms from objects of formula_3 to objects of formula_2. The sets in the formal definition above are the hom-sets between objects of formula_3 and objects of formula_2. (These are also known as het-sets, since the corresponding morphisms can be called "heteromorphisms".) The previous definition can be recovered by the restriction of the hom-functor formula_31 to formula_32.
This also makes it clear that a profunctor can be thought of as a relation between the objects of formula_2 and the objects of formula_3, where each member of the relation is associated with a set of morphisms. A functor is a special case of a profunctor in the same way that a function is a special case of a relation.
Composition of profunctors.
The composite formula_35 of two profunctors
is given by
where formula_39 is the left Kan extension of the functor formula_40 along the Yoneda functor formula_41 of formula_3 (which to every object formula_43 of formula_3 associates the functor formula_45).
It can be shown that
where formula_47 is the least equivalence relation such that formula_48 whenever there exists a morphism formula_49 in formula_3 such that
The bicategory of profunctors.
Composition of profunctors is associative only up to isomorphism (because the product is not strictly associative in Set). The best one can hope is therefore to build a bicategory Prof whose
Properties.
Lifting functors to profunctors.
A functor formula_53 can be seen as a profunctor formula_54 by postcomposing with the Yoneda functor:
It can be shown that such a profunctor formula_56 has a right adjoint. Moreover, this is a characterization: a profunctor formula_4 has a right adjoint if and only if formula_58 factors through the Cauchy completion of formula_3, i.e. there exists a functor formula_53 such that formula_61.


Isotopy of loops

In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop.
Isotopy for loops and quasigroups was introduced by , based on his slightly earlier definition of isotopy for algebras, which was in turn inspired by work of Steenrod.
Isotopy of quasigroups.
Each quasigroup is isotopic to a loop.
Let formula_1 and formula_2 be quasigroups. A quasigroup homotopy from "Q" to "P" is a triple of maps from "Q" to "P" such that
for all "x", "y" in "Q". A quasigroup homomorphism is just a homotopy for which the three maps are equal.
An isotopy is a homotopy for which each of the three maps is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy is given by a permutation of rows "α", a permutation of columns "β", and a permutation on the underlying element set "γ".
An autotopy is an isotopy from a quasigroup formula_1 to itself. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup.
A principal isotopy is an isotopy for which "γ" is the identity map on "Q". In this case the underlying sets of the quasigroups must be the same but the multiplications may differ.
Isotopy of loops.
Let formula_5 and formula_6 be loops and let formula_7 be an isotopy. Then it is the product of the principal isotopy formula_8 from formula_5 and formula_10 and the isomorphism formula_11 between formula_10 and formula_6. Indeed, put formula_14, formula_15 and define the operation ∗ by formula_16.
Let formula_5 and formula_18 be loops and let "e" be the neutral element of formula_5. Let formula_20 a principal isotopy from formula_5 to formula_18. Then formula_23 and formula_24 where formula_25 and formula_26.
A loop "L" is a G-loop if it is isomorphic to all its loop isotopes.
Pseudo-automorphisms of loops.
Let "L" be a loop and "c" an element of "L". A bijection "α" of "L" is called a right pseudo-automorphism of "L" with companion element "c" if for all "x", "y" the identity
holds. One defines left pseudo-automorphisms analogously.
Universal properties.
We say that a loop property "P" is universal if it is isotopy invariant, that is, "P" holds for a loop "L" if and only if "P" holds for all loop isotopes of "L". Clearly, it is enough to check if "P" holds for all principal isotopes of "L".
For example, since the isotopes of a commutative loop need not be commutative, commutativity is not universal. However, associativity and being an abelian group are universal properties. In fact, every group is a G-loop.
The geometric interpretation of isotopy.
Given a loop "L", one can define an incidence geometric structure called a 3-net. Conversely, after fixing an origin and an order of the line classes, a 3-net gives rise to a loop. Choosing a different origin or exchanging the line classes may result in nonisomorphic coordinate loops. However, the coordinate loops are always isotopic. In other words, two loops are isotopic if and only if they are equivalent from "geometric point of view".
The dictionary between algebraic and geometric concepts is as follows


Ilya Kapovich

Ilya Kapovich is a Russian-American mathematician and a Professor of Mathematics at the Hunter College of CUNY. He is known for his contributions to geometric group theory, geometric topology, and complexity theory.
Career.
He received his Ph.D. from the Graduate Center of CUNY in 1996 under the direction of Gilbert Baumslag. After postdoc positions at Rutgers University and Hebrew University, he worked at the University of Illinois at Urbana Champaign from 2000 to 2018. While at Illinois, he has had several visiting positions. Most notably, he was an Alexander von Humboldt Foundation research fellow at Frankfurt University during Spring 2006, and Ada Peluso Visiting Professor at Hunter College during Spring 2017.
Recognition.
In 2012, he was named an inaugural fellow of the American Mathematical Society. He delivered an invited address at 2008 Spring Eastern Sectional meeting of the AMS.
Contributions.
He has worked on problems related to automorphisms of free groups, free-by-cyclic groups, mapping class groups as well as complexity and decision problems in group theory. He has supervised 8 PhD students to completion as of 2020.
Between 2010 and 2015, he served on the editorial board of the LMS Journal of Mathematics and Computation.
His older brother Michael Kapovich is a professor of mathematics at the University of California, Davis, and his twin brother Vitali Kapovitch is a professor of mathematics at the University of Toronto.


Jacques Philippe Marie Binet

Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley and others. In his memoir on the theory of the conjugate axis and of the moment of inertia of bodies he enumerated the principle now known as "Binet's theorem". He is also recognized as the first to describe the rule for multiplying matrices in 1812, and Binet's Formula expressing Fibonacci numbers in closed form is named in his honour, although the same result was known to Abraham de Moivre a century earlier.
Career.
Binet graduated from l'École Polytechnique in 1806, and returned as a teacher in 1807. He advanced in position until 1816 when he became an inspector of studies at l'École. He held this post until 13 November 1830, when he was dismissed by the recently crowned King Louis-Philippe of France, probably because of Binet's strong support of the previous King, Charles X. In 1823 Binet succeeded Delambre in the chair of astronomy at the "Collège de France". He was made a Chevalier in the Légion d'Honneur in 1821, and was elected to the Académie des Sciences in 1843.
Binet's Fibonacci number formula.
This formula provides the formula_1 term in the Fibonacci sequence, and is defined using the recurrence formula:
where


Modular representation theory

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field "K" of positive characteristic "p", necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were too small in an appropriate sense. Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program.
If the characteristic "p" of "K" does not divide the order |"G"|, then modular representations are completely reducible, as with "ordinary" (characteristic 0) representations, by virtue of Maschke's theorem. In the other case, when |"G"| ≡ 0 mod "p", the process of averaging over the group needed to prove Maschke's theorem breaks down, and representations need not be completely reducible. Much of the discussion below implicitly assumes that the field "K" is sufficiently large (for example, "K" algebraically closed suffices), otherwise some statements need refinement.
History.
The earliest work on representation theory over finite fields is by who showed that when "p" does not divide the order of the group, the representation theory is similar to that in characteristic 0. He also investigated modular invariants of some finite groups. The systematic study of modular representations, when the characteristic "p" divides the order of the group, was started by and was continued by him for the next few decades.
Example.
Finding a representation of the cyclic group of two elements over F2 is equivalent to the problem of finding matrices whose square is the identity matrix. Over every field of characteristic other than 2, there is always a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as
Over F2, there are many other possible matrices, such as
Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the Jordan normal form. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.
Ring theory interpretation.
Given a field "K" and a finite group "G", the group algebra "K"["G"] (which is the "K"-vector space with "K"-basis consisting of the elements of "G", endowed with algebra multiplication by extending the multiplication of "G" by linearity) is an Artinian ring.
When the order of "G" is divisible by the characteristic of "K", the group algebra is not semisimple, hence has non-zero Jacobson radical. In that case, there are finite-dimensional modules for the group algebra that are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective.
Brauer characters.
Modular representation theory was developed by Richard Brauer from about 1940 onwards to study in greater depth the relationships between the
characteristic "p" representation theory, ordinary character theory and structure of "G", especially as the latter relates to the embedding of, and relationships between, its "p"-subgroups. Such results can be applied in group theory to problems not directly phrased in terms of representations.
Brauer introduced the notion now known as the Brauer character. When "K" is algebraically closed of positive characteristic "p", there is a bijection between roots of unity in "K" and complex roots of unity of order prime to "p". Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to "p" the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation.
The Brauer character of a representation determines its composition
factors but not, in general, its equivalence type. The irreducible
Brauer characters are those afforded by the simple modules.
These are integral (though not necessarily non-negative) combinations
of the restrictions to elements of order coprime to "p" of the ordinary irreducible
characters. Conversely, the restriction to the elements of order coprime to "p" of
each ordinary irreducible character is uniquely expressible as a non-negative
integer combination of irreducible Brauer characters.
Reduction (mod "p").
In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the
group algebra of the group "G" over a complete discrete
valuation ring "R" with residue field "K" of positive
characteristic "p" and field of fractions "F" of characteristic
0, such as the "p"-adic integers. The structure of "R"["G"] is closely related both to
the structure of the group algebra "K"["G"] and to the structure of the semisimple group algebra "F"["G"], and there is much interplay
between the module theory of the three algebras.
Each "R"["G"]-module naturally gives rise to an "F"["G"]-module,
and, by a process often known informally as reduction (mod "p"),
to a "K"["G"]-module. On the other hand, since "R" is a
principal ideal domain, each finite-dimensional "F"["G"]-module
arises by extension of scalars from an "R"["G"]-module. In general,
however, not all "K"["G"]-modules arise as reductions (mod "p") of
"R"["G"]-modules. Those that do are liftable.
Number of simple modules.
In ordinary representation theory, the number of simple modules "k"("G") is equal to the number of conjugacy classes of "G". In the modular case, the number "l"("G") of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime "p", the so-called "p"-regular classes.
Blocks and the structure of the group algebra.
In modular representation theory, while Maschke's theorem does not hold
when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as blocks. When the field "F" has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra "F"["G"] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when "F" is sufficiently large: each block is a full matrix algebra over "F", the endomorphism ring of the vector space underlying the associated simple module.
To obtain the blocks, the identity element of the group "G" is decomposed as a sum of primitive idempotents
in "Z"("R"[G]), the center of the group algebra over the maximal order "R" of "F". The block corresponding to the primitive idempotent
"e" is the two-sided ideal "e" "R"["G"]. For each indecomposable "R"["G"]-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its composition factors also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the trivial module is known as the principal block.
Projective modules.
In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules with characteristic dividing the group order are rarely projective. Indeed, if a simple module is projective, then it is the only simple module in its block, which is then isomorphic to the endomorphism algebra of the underlying vector space, a full matrix algebra. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine.
For the group algebra of a finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the socle of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomposables have
non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module).
Each projective indecomposable module (and hence each projective module) in positive characteristic "p" may be lifted to a module in characteristic 0. Using the ring "R" as above, with residue field "K", the identity element of "G" may be decomposed as a sum of mutually orthogonal primitive idempotents (not necessarily
central) of "K"["G"]. Each projective indecomposable "K"["G"]-module is isomorphic to "e"."K"["G"] for a primitive idempotent "e" that occurs in this decomposition. The idempotent "e" lifts to a primitive idempotent, say "E", of "R"["G"], and the left module "E"."R"["G"] has reduction (mod "p") isomorphic to "e"."K"["G"].
Some orthogonality relations for Brauer characters.
When a projective module is lifted, the associated character vanishes on all elements of order divisible by "p", and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic "p" module on "p"-regular elements. The (usual character-ring) inner product of the Brauer character of a projective indecomposable with any other Brauer character can thus be defined: this is 0 if the
second Brauer character is that of the socle of a non-isomorphic projective indecomposable, and 1
if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreducible
character in the character of the lift of a projective indecomposable is equal to the number
of occurrences of the Brauer character of the socle of the projective indecomposable when the restriction of the ordinary character to "p"-regular elements is expressed as a sum of irreducible Brauer characters.
Decomposition matrix and Cartan matrix.
The composition factors of the projective indecomposable modules may be calculated as follows:
Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters. The integers involved can be placed in a matrix, with the ordinary irreducible characters assigned rows and the irreducible Brauer characters assigned columns. This is referred to as the "decomposition matrix", and is frequently labelled "D". It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively. The product of the transpose of "D" with "D" itself
results in the Cartan matrix, usually denoted "C"; this is a symmetric matrix such that the entries in its "j"-th row are the multiplicities of the respective simple modules as composition
factors of the "j"-th projective indecomposable module. The Cartan
matrix is non-singular; in fact, its determinant is a power of the
characteristic of "K".
Since a projective indecomposable module in a given block has
all its composition factors in that same block, each block has
its own Cartan matrix.
Defect groups.
To each block "B" of the group algebra "K"["G"], Brauer associated a certain "p"-subgroup, known as its defect group (where "p" is the characteristic of "K"). Formally, it is the largest "p"-subgroup
"D" of "G" for which there is a Brauer correspondent of "B" for the
subgroup formula_3, where formula_4 is the centralizer of "D" in "G".
The defect group of a block is unique up to conjugacy and has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and Brauer irreducible characters agree on elements of order prime to the relevant characteristic "p", and the simple module is projective. At the other extreme, when "K" has characteristic "p", the Sylow "p"-subgroup of the finite group "G" is a defect group for the principal block of "K"["G"].
The order of the defect group of a block has many arithmetical characterizations related to representation theory. It is the largest invariant factor of the Cartan matrix of the block, and occurs with
multiplicity one. Also, the power of "p" dividing the index of the defect group of a block is the greatest common divisor of the powers of "p" dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of "p" dividing the degrees of the ordinary irreducible characters in that block.
Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the "p"-part of a group element "g" is in the defect group of a given block, then each irreducible character in that block vanishes at "g". This is one of many consequences of Brauer's second main theorem.
The defect group of a block also has several characterizations in the more module-theoretic approach to block theory, building on the work of J. A. Green, which associates a "p"-subgroup
known as the vertex to an indecomposable module, defined in terms of relative projectivity of the module. For example, the vertex of each indecomposable module in a block is contained (up to conjugacy)
in the defect group of the block, and no proper subgroup of the defect group has that property.
Brauer's first main theorem states that the number of blocks of a finite group that have a given "p"-subgroup as defect group is the same as the corresponding number for the normalizer in the group of that "p"-subgroup.
The easiest block structure to analyse with non-trivial defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, E.C. Dade, J.A. Green and J.G. Thompson, among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block.
Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a dihedral group, semidihedral group or (generalized) quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann. The indecomposable modules in wild blocks are extremely difficult to classify, even in principle.


Rigid category

In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual "X"* (the internal Hom "[X, 1]") and a morphism 1 → "X" ⊗ "X"* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexandre Grothendieck) by Neantro Saavedra-Rivano in his thesis on Tannakian categories.
Definition.
There are at least two equivalent definitions of a rigidity.
formula_3
formula_4
are identities. A right rigid object is defined similarly.
An inverse is an object "X−1" such that both "X" ⊗ "X−1" and "X−1" ⊗ "X" are isomorphic to 1, the one object of the monoidal category. If an object "X" has a left (resp. right) inverse "X"−1 with respect to the tensor product then it is left (resp. right) rigid, and "X"* = "X"−1.
The operation of taking duals gives a contravariant functor on a rigid category.
Uses.
One important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any rigid category such that taking the "( )"**, the functor of taking the dual twice repeated, is isomorphic to the identity functor, i.e. a pivotal category. Then for any right rigid object "X", and any other object "Y", we may define the isomorphism
formula_5
and its reciprocal isomorphism
formula_6.
Then for any endomorphism formula_7, the trace is of "f" is defined as the composition:
formula_8
We may continue further and define the dimension of a rigid object to be:
formula_9.
Rigidity is also important because of its relation to internal Hom's. If "X" is a left rigid object, then every internal Hom of the form "[X, Z]" exists and is isomorphic to "Z" ⊗ "Y". In particular, in a rigid category, all internal Hom's exist.
Alternative terminology.
A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.
Discussion.
A monoidal category is a category with a tensor product, precisely the sort of category for which rigidity makes sense.


Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.
"Category theory" is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.
In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages.
Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two "different" categories may also be considered "equivalent" for purposes of category theory, even if they do not have precisely the same structure.
Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set, the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the category of topological spaces and continuous maps. All of the preceding categories have the identity map as identity arrows and composition as the associative operation on arrows.
The classic and still much used text on category theory is "Categories for the Working Mathematician" by Saunders Mac Lane. Other references are given in the References below. The basic definitions in this article are contained within the first few chapters of any of these books.
Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.
Definition.
There are many equivalent definitions of a category. One commonly used definition is as follows. A category "C" consists of
Note: Here hom("a", "b") denotes the subclass of morphisms "f" in hom("C") such that formula_3 and formula_4. Such morphisms are often written as "f" : "a" → "b".
such that the following axioms hold:
We write "f": "a" → "b", and we say ""f" is a morphism from "a" to "b"". We write hom("a", "b") (or hom"C"("a", "b") when there may be confusion about to which category hom("a", "b") refers) to denote the hom-class of all morphisms from "a" to "b". From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.
Small and large categories.
A category "C" is called small if both ob("C") and hom("C") are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects "a" and "b", the hom-class hom("a", "b") is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. Since, in small categories, the objects form a set, a small category can be viewed as an algebraic structure similar to a monoid but without requiring closure properties. Large categories on the other hand can be used to create "structures" of algebraic structures.
Examples.
The class of all sets (as objects) together with all functions between them (as morphisms), where the composition of morphisms is the usual function composition, forms a large category, Set. It is the most basic and the most commonly used category in mathematics. The category Rel consists of all sets (as objects) with binary relations between them (as morphisms). Abstracting from relations instead of functions yields allegories, a special class of categories.
Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called discrete. For any given set "I", the "discrete category on I" is the small category that has the elements of "I" as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category.
Any preordered set ("P", ≤) forms a small category, where the objects are the members of "P", the morphisms are arrows pointing from "x" to "y" when "x" ≤ "y". Furthermore, if "≤" is antisymmetric, there can be at most one morphism between any two objects. The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity and the transitivity of the preorder. By the same argument, any partially ordered set and any equivalence relation can be seen as a small category. Any ordinal number can be seen as a category when viewed as an ordered set.
Any monoid (any algebraic structure with a single associative binary operation and an identity element) forms a small category with a single object "x". (Here, "x" is any fixed set.) The morphisms from "x" to "x" are precisely the elements of the monoid, the identity morphism of "x" is the identity of the monoid, and the categorical composition of morphisms is given by the monoid operation. Several definitions and theorems about monoids may be generalized for categories.
Similarly any group can be seen as a category with a single object in which every morphism is "invertible", that is, for every morphism "f" there is a morphism "g" that is both left and right inverse to "f" under composition. A morphism that is invertible in this sense is called an isomorphism.
A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations. Actually, in the view of category the only difference between groupoid and group is that a groupoid may have more than one object but the group must have only one. Consider a topological space "X" and fix a base point formula_5 of "X", then formula_6 is the fundamental group of the topological space "X" and the base point formula_5, and as a set it has the structure of group; if then let the base point formula_5 runs over all points of "X", and take the union of all formula_6, then the set we get has only the structure of groupoid (which is called as the fundamental groupoid of "X"): two loops (under equivalence relation of homotopy) may not have the same base point so they can not multiple with each other. In the language of category, this means here two morphisms may not have the same source object (or target object, because in this case for any morphism the source object and the target object are same: the base point) so they can not compose with each other.
Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths. Such a category is called the "free category" generated by the graph.
The class of all preordered sets with monotonic functions as morphisms forms a category, Ord. It is a concrete category, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure.
The class of all groups with group homomorphisms as morphisms and function composition as the composition operation forms a large category, Grp. Like Ord, Grp is a concrete category. The category Ab, consisting of all abelian groups and their group homomorphisms, is a full subcategory of Grp, and the prototype of an abelian category. Other examples of concrete categories are given by the following table.
Fiber bundles with bundle maps between them form a concrete category.
The category Cat consists of all small categories, with functors between them as morphisms.
Construction of new categories.
Dual category.
Any category "C" can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the "dual" or "opposite category" and is denoted "C"op.
Product categories.
If "C" and "D" are categories, one can form the "product category" "C" × "D": the objects are pairs consisting of one object from "C" and one from "D", and the morphisms are also pairs, consisting of one morphism in "C" and one in "D". Such pairs can be composed componentwise.
Types of morphisms.
A morphism "f" : "a" → "b" is called
Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:
Relations among morphisms (such as "fg" = "h") can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.


Shou-Wu Zhang

Shou-Wu Zhang (; born October 9, 1962) is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.
Biography.
Early life.
Shou-Wu Zhang was born in Hexian, Ma'anshan, Anhui, China on October 9, 1962. Zhang grew up in a poor farming household and could not attend school until eighth grade due to the Cultural Revolution. He spent most of his childhood raising ducks in the countryside and self-studying mathematics textbooks that he acquired from sent-down youth in trades for frogs. By the time he entered junior high school at the age of fourteen, he had self-learned calculus and had become interested in number theory after reading about Chen Jingrun's proof of Chen's theorem which made substantial progress on Goldbach's conjecture.
Education.
Zhang was admitted to the Sun Yat-sen University chemistry department in 1980 after scoring poorly on his mathematics entrance examinations, but he later transferred to the mathematics department after feigning color blindness and received his bachelor's degree in mathematics in 1983. He then studied under analytic number theorist Wang Yuan at the Chinese Academy of Sciences where he received his master's degree in 1986. In 1986, Zhang was brought to the United States to pursue his doctoral studies at Columbia University by Dorian M. Goldfeld. He then studied under Goldfeld, Hervé Jacquet, Lucien Szpiro, and Gerd Faltings, and then completed his PhD at Columbia University under Szpiro in 1991.
Career.
Zhang was a member of the Institute for Advanced Study and an assistant professor at Princeton University from 1991 to 1996. In 1996, Zhang moved back to Columbia University where he was a tenured professor until 2013. He has been a professor at Princeton University since 2011 and was the Henry Burchard Fine Professor from 2014 to 2015.
Zhang is on the editorial boards of: "Acta Mathematica Sinica", "Algebra & Number Theory", "Forum of Mathematics", "Journal of Differential Geometry", "National Science Review", "Pure and Applied Mathematics Quarterly", "Science in China", and "Research in Number Theory". He has previously served on the editorial boards of: "Journal of Number Theory", "Journal of the American Mathematical Society", "Journal of Algebraic Geometry", and "International Journal of Number Theory".
Research.
Zhang's doctoral thesis "Positive line bundles on Arithmetic Surfaces" proved a Nakai–Moishezon type theorem in intersection theory using a result from differential geometry already proved in Tian Gang's doctoral thesis. In a series of subsequent papers (, ), he further developed his theory of 'positive line bundles' in Arakelov theory which culminated in a proof (with Emmanuel Ullmo) of the Bogomolov conjecture ().
In a series of works in the 2000s (, ), Zhang proved a generalization of the Gross–Zagier theorem from elliptic curves over rationals to modular abelian varieties of GL(2) type over totally real fields. In particular, the latter result led him to a proof of the rank one Birch-Swinnerton-Dyer conjecture for modular abelian varieties of GL(2) type over totally real fields through his work relating the Néron–Tate height of Heegner points to special values of L-functions in . Eventually, established a full generalization of the Gross–Zagier theorem to all Shimura curves.
In arithmetic dynamics, posed conjectures on the Zariski density of non-fibered endomorphisms of quasi-projective varieties and proposed a dynamical analogue of the Manin–Mumford conjecture.
In 2018, proved the averaged Colmez conjecture which was shown to imply the André–Oort conjecture for Siegel modular varieties by Jacob Tsimerman.
Awards.
Zhang has received a Sloan Foundation Research Fellowship (1997) and a Morningside Gold Medal of Mathematics (1998). He is also a Clay Foundation Prize Fellow (2003), Guggenheim Foundation Fellow (2009), Fellow of the American Academy of Arts and Sciences (2011), and Fellow of the American Mathematical Society (2016). He was also an invited speaker at the International Congress of Mathematicians in 1998.


Reciprocal polynomial

In algebra, given a polynomial
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by or , is the polynomial
That is, the coefficients of are the coefficients of in reverse order. They arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix.
In the special case where the field is the complex numbers, when
the conjugate reciprocal polynomial, denoted , is defined by,
where formula_5 denotes the complex conjugate of formula_6, and is also called the reciprocal polynomial when no confusion can arise.
A polynomial is called self-reciprocal or palindromic if .
The coefficients of a self-reciprocal polynomial satisfy for all . In the conjugate reciprocal case, the coefficients must be real to satisfy the condition.
Properties.
Reciprocal polynomials have several connections with their original polynomials, including:
Other properties of reciprocal polynomials may be obtained, for instance:
Palindromic and antipalindromic polynomials.
A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if
is a polynomial of degree , then is "palindromic" if for . Some authors use the terms "palindromic" and "reciprocal" interchangeably.
Similarly, a polynomial of degree is called antipalindromic if for . That is, a polynomial is "antipalindromic" if .
Examples.
From the properties of the binomial coefficients, it follows that the polynomials are palindromic for all positive integers , while the polynomials are palindromic when is even and antipalindromic when is odd.
Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
Real coefficients.
A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane (that is, all the roots have modulus 1) is either palindromic or antipalindromic.
Conjugate reciprocal polynomials.
A polynomial is conjugate reciprocal if formula_8 and self-inversive if formula_9 for a scale factor on the unit circle.
If is the minimal polynomial of with , and has real coefficients, then is self-reciprocal. This follows because
So is a root of the polynomial formula_11 which has degree . But, the minimal polynomial is unique, hence
for some constant , i.e. formula_13. Sum from to and note that 1 is not a root of . We conclude that .
A consequence is that the cyclotomic polynomials are self-reciprocal for . This is used in the special number field sieve to allow numbers of the form and to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that (Euler's totient function) of the exponents are 10, 12, 8 and 12.
Application in coding theory.
The reciprocal polynomial finds a use in the theory of cyclic error correcting codes. Suppose can be factored into the product of two polynomials, say . When generates a cyclic code , then the reciprocal polynomial generates , the orthogonal complement of .
Also, is "self-orthogonal" (that is, , if and only if divides .


Kronecker product

In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product.
The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the "Zehfuss matrix", and the "Zehfuss product", after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used.
Definition.
If A is an matrix and B is a matrix, then the Kronecker product is the block matrix:
more explicitly:
Using formula_3 and formula_4 to denote truncating integer division and remainder, respectively, and numbering the matrix elements starting from 0, one obtains formula_5 and formula_6 For the usual numbering starting from 1, one obtains
formula_7
and
formula_8
If A and B represent linear transformations and , respectively, then represents the tensor product of the two maps, .
Examples.
Similarly:
Properties.
Relations to other matrix operations.
) = \exp(\mathbf{N}) \otimes \exp(\mathbf{M}) </math>
Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let "Hi" be the Hamiltonian of the "i"th such system. Then the total Hamiltonian of the ensemble is
Matrix equations.
The Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation , where A, B and C are given matrices and the matrix X is the unknown. We can use the "vec trick" to rewrite this equation as
Here, vec(X) denotes the vectorization of the matrix X, formed by stacking the columns of X into a single column vector.
It now follows from the properties of the Kronecker product that the equation has a unique solution, if and only if A and B are invertible .
If X and C are row-ordered into the column vectors u and v, respectively, then
The reason is that
Applications.
For an example of the application of this formula, see the article on the Lyapunov equation. This formula also comes in handy in showing that the matrix normal distribution is a special case of the multivariate normal distribution. This formula is also useful for representing 2D image processing operations in matrix-vector form.
Another example is when a matrix can be factored as a Hadamard product, then matrix multiplication can be performed faster by using the above formula. This can be applied recursively, as done in the radix-2 FFT and the Fast Walsh–Hadamard transform. Splitting a known matrix into the Hadamard product of two smaller matrices is known as the "nearest Kronecker product" problem, and can be solved exactly by using the SVD. To split a matrix into the Hadamard product of more than two matrices, in an optimal fashion, is a difficult problem and the subject of ongoing research; some authors cast it as a tensor decomposition problem.
In conjunction with the least squares method, the Kronecker product can be used as an accurate solution to the hand eye calibration problem.
Related matrix operations.
Two related matrix operations are the Tracy–Singh and Khatri–Rao products, which operate on partitioned matrices. Let the matrix A be partitioned into the blocks A"ij" and matrix B into the blocks B"kl", with of course , , and .
Tracy–Singh product.
The Tracy–Singh product is defined as
which means that the ("ij")-th subblock of the product is the matrix , of which the ("k")-th subblock equals the matrix . Essentially the Tracy–Singh product is the pairwise Kronecker product for each pair of partitions in the two matrices.
For example, if A and B both are partitioned matrices e.g.:
we get:
Face-splitting product.
Mixed-products properties
where formula_19 denotes the Face-splitting product
Similarly:
where formula_23 and formula_24 are vectors,
where formula_23 and formula_24 are vectors, formula_28 denotes the Hadamard product
Similarly:
where formula_31 is vector convolution and formula_32 is the Fourier transform matrix (this result is an evolving of count sketch properties),
where formula_34 denotes the Column-wise Khatri–Rao product.
Similarly:
where formula_23 and formula_24 are vectors


External (mathematics)

The term external is useful for describing certain algebraic structures. The term comes from the concept of an external binary operation which is a binary operation that draws from some "external set". To be more specific, a left external binary operation on "S" over "R" is a function formula_1 and a right external binary operation on "S" over "R" is a function formula_2 where "S" is the set the operation is defined on, and "R" is the external set (the set the operation is defined "over").
Generalizations.
The "external" concept is a generalization rather than a specialization, and as such, it is different from many terms in mathematics. A similar but opposite concept is that of an "internal binary function" from "R" to "S", defined as a function formula_3. Internal binary functions are like binary functions, but are a form of specialization, so they only accept a subset of the domains of binary functions. Here we list these terms with the function signatures they imply, along with some examples:
External monoids.
Since monoids are defined in terms of binary operations, we can define an "external monoid" in terms of "external binary operations". For the sake of simplicity, unless otherwise specified, a "left" external binary operation is implied. Using the term "external", we can make the generalizations:
Modules as external rings.
Much of the machinery of modules and vector spaces are fairly straightforward, or discussed above. The only thing not covered yet is their distribution axioms. The external ring multiplication formula_25 is externally distributive in formula_26 over the ring formula_27 iff:
Using these terminology we can make the following local generalizations:
Other examples.
Now that we have all the terminology we need, we can make simple connections between various structures:
Usefulness.
It could be argued that we already have terms for the concepts described here, like dynamical systems, group actions, modules, and vector spaces. However, there is still no other terminology available for an external monoid for which this terminology gives us a concise expression. Above all else, this is a reason this term should be of use in the mathematical community.


Schur product theorem

In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in "Journal für die reine und angewandte Mathematik".)
Proof.
Proof using the trace formula.
For any matrices formula_1 and formula_2, the Hadamard product formula_3 considered as a bilinear form acts on vectors formula_4 as
where formula_6 is the matrix trace and formula_7 is the diagonal matrix having as diagonal entries the elements of formula_8.
Suppose formula_1 and formula_2 are positive definite, and so Hermitian. We can consider their square-roots formula_11 and formula_12, which are also Hermitian, and write
Then, for formula_14, this is written as formula_15 for formula_16 and thus is strictly positive for formula_17, which occurs if and only if formula_18. This shows that formula_19 is a positive definite matrix.
Proof using Gaussian integration.
Case of "M" = "N".
Let formula_20 be an formula_21-dimensional centered Gaussian random variable with covariance formula_22. Then the covariance matrix of formula_23 and formula_24 is
Using Wick's theorem to develop formula_26 we have
Since a covariance matrix is positive definite, this proves that the matrix with elements formula_28 is a positive definite matrix.
General case.
Let formula_20 and formula_30 be formula_21-dimensional centered Gaussian random variables with covariances formula_32, formula_33 and independent from each other so that we have
Then the covariance matrix of formula_36 and formula_37 is
Using Wick's theorem to develop
and also using the independence of formula_20 and formula_30, we have
Since a covariance matrix is positive definite, this proves that the matrix with elements formula_43 is a positive definite matrix.
Proof using eigendecomposition.
Proof of positive semidefiniteness.
Let formula_44 and formula_45. Then
Each formula_47 is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, formula_48 thus the sum formula_3 is also positive semidefinite.
Proof of definiteness.
To show that the result is positive definite requires further proof. We shall show that for any vector formula_18, we have formula_51. Continuing as above, each formula_52, so it remains to show that there exist formula_53 and formula_54 for which corresponding term above is nonzero. For this we observe that
Since formula_2 is positive definite, there is a formula_54 for which formula_58 (since otherwise formula_59 for all formula_54), and likewise since formula_1 is positive definite there exists an formula_53 for which formula_63 However, this last sum is just formula_64. Thus its square is positive. This completes the proof.


Inder Bir Singh Passi

Inder Bir Singh Passi (I.B.S. Passi) is an Indian mathematician who specialises in algebra.
He was awarded in 1983 the Shanti Swarup Bhatnagar Prize for Science and Technology, the highest science award in India, in the mathematical sciences category. Prof. Passi is a noted group-theorist in India, has made significant contribution to certain aspects of theory of groups specially to the study of group rings. His results on the dimension subgroups, augmentation powers in group rings, and related problems have received wide recognition. His 1979 monograph summarizing the state of the subject is a basic reference source.


Kummer theory

In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of "n"th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer "n" – and therefore belong to abstract algebra. The theory of cyclic extensions of the field "K" when the characteristic of "K" does divide "n" is called Artin–Schreier theory.
Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is something much more serious.
Kummer extensions.
A Kummer extension is a field extension "L"/"K", where for some given integer "n" > 1 we have
For example, when "n" = 2, the first condition is always true if "K" has characteristic ≠ 2. The Kummer extensions in this case include quadratic extensions formula_1 where "a" in "K" is a non-square element. By the usual solution of quadratic equations, any extension of degree 2 of "K" has this form. The Kummer extensions in this case also include biquadratic extensions and more general multiquadratic extensions. When "K" has characteristic 2, there are no such Kummer extensions.
Taking "n" = 3, there are no degree 3 Kummer extensions of the rational number field Q, since for three cube roots of 1 complex numbers are required. If one takes "L" to be the splitting field of "X"3 − "a" over Q, where "a" is not a cube in the rational numbers, then "L" contains a subfield "K" with three cube roots of 1; that is because if α and β are roots of the cubic polynomial, we shall have (α/β)3 =1 and the cubic is a separable polynomial. Then "L"/"K" is a Kummer extension.
More generally, it is true that when "K" contains "n" distinct "n"th roots of unity, which implies that the characteristic of "K" doesn't divide "n", then adjoining to "K" the "n"th root of any element "a" of "K" creates a Kummer extension (of degree "m", for some "m" dividing "n"). As the splitting field of the polynomial "Xn" − "a", the Kummer extension is necessarily Galois, with Galois group that is cyclic of order "m". It is easy to track the Galois action via the root of unity in front of formula_2
Kummer theory provides converse statements. When "K" contains "n" distinct "n"th roots of unity, it states that any abelian extension of "K" of exponent dividing "n" is formed by extraction of roots of elements of "K". Further, if "K"× denotes the multiplicative group of non-zero elements of "K", abelian extensions of "K" of exponent "n" correspond bijectively with subgroups of
that is, elements of "K"× modulo "n"th powers. The correspondence can be described explicitly as follows. Given a subgroup
the corresponding extension is given by
where
In fact it suffices to adjoin "n"th root of one representative of each element of any set of generators of the group Δ. Conversely, if "L" is a Kummer extension of "K", then Δ is recovered by the rule
In this case there is an isomorphism
given by
where α is any "n"th root of "a" in "L". Here formula_10 denotes the multiplicative group of "n"th roots of unity (which belong to "K") and formula_11 is the group of continuous homomorphisms from formula_12 equipped with Krull topology to formula_10 with discrete topology (with group operation given by pointwise multiplication). This group (with discrete topology) can also be viewed as Pontryagin dual of formula_12, assuming we regard formula_10 as a subgroup of circle group. If the extension "L"/"K" is finite, then formula_12 is a finite discrete group and we have
however the last isomorphism isn't natural.
Recovering formula_18 from a primitive element.
For formula_19 prime, let formula_20 be a field containing formula_21 and formula_22 a degree formula_19 Galois extension. Note the Galois group is cyclic, generated by formula_24. Let
Then
Since formula_27 and
When formula_29 is an abelian extension of degree formula_30 square-free such that formula_31, apply the same argument to the subfields formula_32 Galois of degree formula_33 to obtain
where
Generalizations.
Suppose that "G" is a profinite group acting on a module "A" with a surjective homomorphism π from the "G"-module "A" to itself. Suppose also that "G" acts trivially on the kernel "C" of π and that the first cohomology group H1("G","A") is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between "A""G"/π("A""G") and Hom("G","C").
Kummer theory is the special case of this when "A" is the multiplicative group of the separable closure of a field "k", "G" is the Galois group, π is the "n"th power map, and "C" the group of "n"th roots of unity. Artin–Schreier theory is the special case when "A" is the additive group of the separable closure of a field "k" of positive characteristic "p", "G" is the Galois group, π is the Frobenius map minus the identity, and "C" the finite field of order "p". Taking "A" to be a ring of truncated Witt vectors gives Witt's generalization of Artin–Schreier theory to extensions of exponent dividing "pn".


Glossary of ring theory

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.
For the items in commutative algebra (the theory of commutative rings), see glossary of commutative algebra. For ring-theoretic concepts in the language of modules, see also Glossary of module theory.
For specific types of algebras, see also: Glossary of field theory and Glossary of Lie groups and Lie algebras. Since, currently, there is no glossary on not-necessarily-associative algebra-structures in general, this glossary includes some concepts that do not need associativity; e.g., a derivation.


T-group (mathematics)

In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups:
The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups "G" with an abelian normal Hall subgroup "H" of odd order such that the quotient group "G"/"H" is a Dedekind group and "H" is acted upon by conjugation as a group of power automorphisms by "G".
A PT-group is a group in which permutability is transitive. A finite T-group is a PT-group.


Carlos Simpson

Carlos Tschudi Simpson (born 30 June 1962) is an American mathematician, specializing in algebraic geometry.
Simpson received his Ph.D. in 1987 from Harvard University, where he was supervised by Wilfried Schmid; his thesis was titled "Systems of Hodge Bundles and Uniformization". He became a professor at the University of Toulouse III (Paul Sabatier University) and then at the University of Nice. He is research director of the Centre national de la recherche scientifique.
He works on moduli spaces of vector bundles, higher non-abelian de Rham cohomology (Hodge theory), the theory of higher categories and computer verification of mathematical proofs ("e.g." verification of proofs within Zermelo–Fraenkel set theory using Coq). In his Ph.D. dissertation, Simpson studied the notion of system of Hodge bundles, which can be seen as a special case of the higher dimensional generalization of Higgs bundles introduced earlier by Nigel Hitchin. The Simpson correspondence (or the Corlette-Simpson correspondence, named after Kevin Corlette and Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, complex algebraic curve.
The Deligne–Simpson Problem, an algebraic problem associated with monodromy matrices, is named after Carlos Simpson and Pierre Deligne.
Simpson was an Invited Speaker with talk "Nonabelian Hodge theory" at the International Congress of Mathematicians in 1990 at Kyoto. In 2015 he received the Sophie Germain Prize.


Eisenstein integral

In mathematical representation theory, the Eisenstein integral is an integral introduced by in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms.
 used Eisenstein integrals to decompose the regular representation of a semisimple Lie group into representations induced from parabolic subgroups.
 gave a survey of Harish-Chandra's work on this.
Definition.
 defined the Eisenstein integral by
where:


Split exact sequence

In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
Equivalent characterizations.
A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an abelian category
is called split exact if it is isomorphic to the sequence where the middle term is the direct sum of the outer ones:
The requirement that the sequence is isomorphic means that there is an isomorphism formula_3 such that the composite formula_4 is the natural inclusion formula_5 and such that the composite formula_6 equals "b". This can be summarized by a commutative diagram as:
The splitting lemma provides further equivalent characterizations of split exact sequences.
Examples.
A trivial example of a split short exact sequence is
where formula_8 are R-modules, formula_9 is the canonical injection and formula_10 is the canonical projection.
Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis.
The exact sequence formula_11 (where the first map is multiplication by 2) is not split exact.
Related notions.
Pure exact sequences can be characterized as the filtered colimits of split exact sequences.


Continuous q-Hahn polynomials

In mathematics, the continuous "q"-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
formula_2


Refinement monoid

In mathematics, a refinement monoid is a commutative monoid "M" such that for any elements "a0", "a1", "b0", "b1" of "M" such that "a0+a1=b0+b1", there are elements "c00", "c01", "c10", "c11" of "M" such that "a0=c00+c01", "a1=c10+c11", "b0=c00+c10", and "b1=c01+c11".
A commutative monoid "M" is said to be conical if "x"+"y"=0 implies that "x"="y"=0, for any elements "x","y" of "M".
Basic examples.
A join-semilattice with zero is a refinement monoid if and only if it is distributive.
Any abelian group is a refinement monoid.
The positive cone "G+" of a partially ordered abelian group "G" is a refinement monoid if and only if "G" is an "interpolation group", the latter meaning that for any elements "a0", "a1", "b0", "b1" of "G" such that "ai ≤ bj" for all "i, j<2", there exists an element "x" of "G" such that "ai ≤ x ≤ bj" for all "i, j<2". This holds, for example, in case "G" is lattice-ordered.
The "isomorphism type" of a Boolean algebra "B" is the class of all Boolean algebras isomorphic to "B". (If we want this to be a set, restrict to Boolean algebras of set-theoretical rank below the one of "B".)
The class of isomorphism types of Boolean algebras, endowed with the addition defined by
formula_1 (for any Boolean algebras "X" and "Y", where formula_2 denotes the isomorphism type of "X"), is a conical refinement monoid.
Vaught measures on Boolean algebras.
For a Boolean algebra "A" and a commutative monoid "M", a map "μ" : "A" → "M" is a "measure", if "μ(a)=0" if and only if "a=0", and "μ(a ∨ b)=μ(a)+μ(b)" whenever "a" and "b" are disjoint (that is, "a ∧ b=0"), for any "a, b" in "A". We say in addition that "μ" is a "Vaught measure" (after Robert Lawson Vaught), or "V-measure", if for all "c" in "A" and all "x,y" in "M" such that "μ(c)=x+y", there are disjoint "a, b" in "A" such that "c=a ∨ b", "μ(a)=x", and "μ(b)=y".
An element "e" in a commutative monoid "M" is "measurable" (with respect to "M"), if there are a Boolean algebra "A" and a V-measure "μ" : "A" → "M" such that "μ(1)=e"---we say that "μ" "measures" "e". We say that "M" is "measurable", if any element of "M" is measurable (with respect to "M"). Of course, every measurable monoid is a conical refinement monoid.
Hans Dobbertin proved in 1983 that any conical refinement monoid with at most ℵ1 elements is measurable. He also proved that any element in an at most countable conical refinement monoid is measured by a unique (up to isomorphism) V-measure on a unique at most countable Boolean algebra.
He raised there the problem whether any conical refinement monoid is measurable. This was answered in the negative by Friedrich Wehrung in 1998. The counterexamples can have any cardinality greater than or equal to ℵ2.
Nonstable K-theory of von Neumann regular rings.
For a ring (with unit) "R", denote by FP("R") the class of finitely generated projective right "R"-modules. Equivalently, the objects of FP("R") are the direct summands of all modules of the form "Rn", with "n" a positive integer, viewed as a right module over itself. Denote by formula_2 the isomorphism type of an object "X" in FP("R"). Then the set "V(R)" of all isomorphism types of members of FP("R"), endowed with the addition defined by formula_4, is a conical commutative monoid. In addition, if "R" is von Neumann regular, then "V(R)" is a refinement monoid. It has the order-unit formula_5. We say that "V(R)" encodes the "nonstable K-theory of R".
For example, if "R" is a division ring, then the members of FP("R") are exactly the finite-dimensional right vector spaces over "R", and two vector spaces are isomorphic if and only if they have the same dimension. Hence "V(R)" is isomorphic to the monoid formula_6 of all natural numbers, endowed with its usual addition.
A slightly more complicated example can be obtained as follows. A "matricial algebra" over a field "F" is a finite product of rings of the form formula_7, the ring of all square matrices with "n" rows and entries in "F", for variable positive integers "n". A direct limit of matricial algebras over "F" is a "locally matricial algebra over F". Every locally matricial algebra is von Neumann regular. For any locally matricial algebra "R", "V(R)" is the positive cone of a so-called "dimension group". By definition, a dimension group is a partially ordered abelian group whose underlying order is directed, whose positive cone is a refinement monoid, and which is "unperforated", the letter meaning that "mx≥0" implies that "x≥0", for any element "x" of "G" and any positive integer "m". Any "simplicial" group, that is, a partially ordered abelian group of the form formula_8, is a dimension group. Effros, Handelman, and Shen proved in 1980 that dimension groups are exactly the direct limits of simplicial groups, where the transition maps are positive homomorphisms. This result had already been proved in 1976, in a slightly different form, by P.A. Grillet. Elliott proved in 1976 that the positive cone of any countable direct limit of simplicial groups is isomorphic to "V(R)", for some locally matricial ring "R". Finally, Goodearl and Handelman proved in 1986 that the positive cone of any dimension group with at most ℵ1 elements is isomorphic to "V(R)", for some locally matricial ring "R" (over any given field).
Wehrung proved in 1998 that there are dimension groups with order-unit whose positive cone cannot be represented as "V(R)", for a von Neumann regular ring "R". The given examples can have any cardinality greater than or equal to ℵ2. Whether any conical refinement monoid with at most ℵ1 (or even ℵ0) elements can be represented as "V(R)" for "R" von Neumann regular is an open problem.


Unitary group

In mathematics, the unitary group of degree "n", denoted U("n"), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.
In the simple case , the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group.
The unitary group U("n") is a real Lie group of dimension "n"2. The Lie algebra of U("n") consists of skew-Hermitian matrices, with the Lie bracket given by the commutator.
The general unitary group (also called the group of unitary similitudes) consists of all matrices "A" such that "A"∗"A" is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.
Properties.
Since the determinant of a unitary matrix is a complex number with norm , the determinant gives a group homomorphism
The kernel of this homomorphism is the set of unitary matrices with determinant . This subgroup is called the special unitary group, denoted . We then have a short exact sequence of Lie groups:
The above map to has a section: we can view as the subgroup of that are diagonal with in the upper left corner and on the rest of the diagonal. Therefore is a semi-direct product of with .
The unitary group is not abelian for . The center of is the set of scalar matrices with ; this follows from Schur's lemma. The center is then isomorphic to . Since the center of is a -dimensional abelian normal subgroup of , the unitary group is not semisimple, but it is reductive.
Topology.
The unitary group U("n") is endowed with the relative topology as a subset of , the set of all complex matrices, which is itself homeomorphic to a 2"n"2-dimensional Euclidean space.
As a topological space, U("n") is both compact and connected. To show that U("n") is connected, recall that any unitary matrix "A" can be diagonalized by another unitary matrix "S". Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write
A path in U("n") from the identity to "A" is then given by
The unitary group is not simply connected; the fundamental group of U("n") is infinite cyclic for all "n":
To see this, note that the above splitting of U("n") as a semidirect product of SU("n") and U(1) induces a topological product structure on U("n"), so that
Now the first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, whereas formula_7 is simply connected.
The determinant map induces an isomorphism of fundamental groups, with the splitting inducing the inverse.
The Weyl group of U("n") is the symmetric group "Sn", acting on the diagonal torus by permuting the entries:
Related groups.
2-out-of-3 property.
The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups:
Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be "compatible" (meaning that one uses the same "J" in the complex structure and the symplectic form, and that this "J" is orthogonal; writing all the groups as matrix groups fixes a "J" (which is orthogonal) and ensures compatibility).
In fact, it is the intersection of any "two" of these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth.
At the level of equations, this can be seen as follows:
Any two of these equations implies the third.
At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). On an almost Kähler manifold, one can write this decomposition as , where is the Hermitian form, is the Riemannian metric, is the almost complex structure, and is the almost symplectic structure.
From the point of view of Lie groups, this can partly be explained as follows: O(2"n") is the maximal compact subgroup of , and U("n") is the maximal compact subgroup of both and Sp(2"n"). Thus the intersection or is the maximal compact subgroup of both of these, so U("n"). From this perspective, what is unexpected is the intersection .
Special unitary and projective unitary groups.
Just as the orthogonal group O("n") has the special orthogonal group SO("n") as subgroup and the projective orthogonal group PO("n") as quotient, and the projective special orthogonal group PSO("n") as subquotient, the unitary group U("n") has associated to it the special unitary group SU("n"), the projective unitary group PU("n"), and the projective special unitary group PSU("n"). These are related as by the commutative diagram at right; notably, both projective groups are equal: .
The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general formula_11.
G-structure: almost Hermitian.
In the language of G-structures, a manifold with a U("n")-structure is an almost Hermitian manifold.
Generalizations.
From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group formula_12, which is an algebraic group that arises from the combination of the "diagram automorphism" of the general linear group (reversing the Dynkin diagram A"n", which corresponds to transpose inverse) and the "field automorphism" of the extension C/R (namely complex conjugation). Both these automorphisms are automorphisms of the algebraic group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group. The classical unitary group is a real form of this group, corresponding to the standard Hermitian form Ψ, which is positive definite.
This can be generalized in a number of ways:
Indefinite forms.
Analogous to the indefinite orthogonal groups, one can define an indefinite unitary group, by considering the transforms that preserve a given Hermitian form, not necessarily positive definite (but generally taken to be non-degenerate). Here one is working with a vector space over the complex numbers.
Given a Hermitian form Ψ on a complex vector space "V", the unitary group U(Ψ) is the group of transforms that preserve the form: the transform "M" such that for all . In terms of matrices, representing the form by a matrix denoted Φ, this says that .
Just as for symmetric forms over the reals, Hermitian forms are determined by signature, and are all unitarily congruent to a diagonal form with "p" entries of 1 on the diagonal and "q" entries of −1. The non-degenerate assumption is equivalent to . In a standard basis, this is represented as a quadratic form as:
and as a symmetric form as:
The resulting group is denoted .
Finite fields.
Over the finite field with elements, F"q", there is a unique quadratic extension field, F"q"2, with order 2 automorphism formula_18 (the "r"th power of the Frobenius automorphism). This allows one to define a Hermitian form on an F"q"2 vector space "V", as an F"q"-bilinear map formula_19 such that formula_20 and formula_21 for . Further, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to
where formula_23 represent the coordinates of in some particular F"q"2-basis of the "n"-dimensional space "V" .
Thus one can define a (unique) unitary group of dimension "n" for the extension F"q"2/F"q", denoted either as or depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted or . For convenience, this article will use the convention. The center of has order and consists of the scalar matrices that are unitary, that is those matrices "cIV" with formula_24. The center of the special unitary group has order and consists of those unitary scalars which also have order dividing "n". The quotient of the unitary group by its center is called the projective unitary group, , and the quotient of the special unitary group by its center is the projective special unitary group . In most cases ( and ), is a perfect group and is a finite simple group, .
Degree-2 separable algebras.
More generally, given a field "k" and a degree-2 separable "k"-algebra "K" (which may be a field extension but need not be), one can define unitary groups with respect to this extension.
First, there is a unique "k"-automorphism of "K" formula_25 which is an involution and fixes exactly "k" (formula_26 if and only if ). This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.
Algebraic groups.
The equations defining a unitary group are polynomial equations over "k" (but not over "K"): for the standard form , the equations are given in matrices as , where formula_27 is the conjugate transpose. Given a different form, they are . The unitary group is thus an algebraic group, whose points over a "k"-algebra "R" are given by:
For the field extension C/R and the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by:
In fact, the unitary group is a linear algebraic group.
Unitary group of a quadratic module.
The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined, which incorporates as special cases many different classical algebraic groups. The definition goes back to Anthony Bak's thesis.
To define it, one has to define quadratic modules first:
Let "R" be a ring with anti-automorphism "J", formula_30 such that formula_31 for all "r" in "R" and formula_32. Define
Let be an additive subgroup of "R", then Λ is called "form parameter" if formula_34 and formula_35. A pair such that "R" is a ring and Λ a form parameter is called "form ring".
Let "M" be an "R"-module and "f" a "J"-sesquilinear form on "M" (i.e., formula_36 for any formula_37 and formula_38). Define formula_39 and formula_40, then "f" is said to "define" the "Λ-quadratic form" on "M". A "quadratic module" over is a triple such that "M" is an "R"-module and is a Λ-quadratic form.
To any quadratic module defined by a "J"-sesquilinear form "f" on "M" over a form ring one can associate the "unitary group"
The special case where , with "J" any non-trivial involution (i.e., formula_42 and gives back the "classical" unitary group (as an algebraic group).
Polynomial invariants.
The unitary groups are the automorphisms of two polynomials in real non-commutative variables:
These are easily seen to be the real and imaginary parts of the complex form formula_44. The two invariants separately are invariants of O(2"n") and Sp(2"n"). Combined they make the invariants of U("n") which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.
Classifying space.
The classifying space for U("n") is described in the article classifying space for U(n).


Ivan Panin (mathematician)

Ivan Aleksandrovich Panin (Иван Александрович Панин, born 2 July 1959 in Apatity, Russia) is a Russian mathematician, specializing in algebra, algebraic geometry, and algebraic K-theory.
Education and career.
In 1973 he entered boarding school at D. K. Fadeev Academic Gymnasium and graduated there in 1976 There he graduated in 1981 from the Faculty of Mathematics and Mechanics of Saint Petersburg State University. At the St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences (abbreviated ПОМИ им. В. А. Стеклова РАН in Russian), he defended in 1984 his thesis for the degree of candidate of physical and mathematical sciences (Ph.D.) with supervisor Andrei Suslin and then became employed there as a staff member. Panin received in 1996 the degree of Doctor nauk from the St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences. There in 1999 he became the head of the laboratory of algebra and number theory. In 2003 he was elected a corresponding member of the Russian Academy of Sciences in the Department of Mathematical Sciences. In 2018 in Rio de Janeiro he was an invited speaker at the International Congress of Mathematicians.
Research.
The main directions of I. A. Panin's work are the theory of oriented cohomology on algebraic varieties, algebraic K-theory of homogeneous varieties, Gersten's conjecture, the Grothendieck-Serre conjecture on principal G-bundles, and purity in algebraic geometry.
I. A. Panin proved (together with A. L. Smirnov) theorems of the Riemann-Roch type for oriented cohomology theories and Riemann-Roch type theorems for the Adams operation. Panin found a proof of Gersten's conjecture in the case of equal characteristic and an affirmative solution (jointly with Manuel Ojanguren) of the "purity" problem for quadratic forms.
Panin computed the algebraic K-groups of all twisted forms of flag varieties and all principal homogeneous spaces over the inner forms of semisimple algebraic groups. He, jointly with A. S. Merkurjev and A. R. Wadsworth, generalized, to arbitrary Borel varieties, results proved by David Tao concerning index reduction formulas for the function fields of involution varieties.


P-residual subgroup


Sylvester domain

In mathematics, a Sylvester domain, named after James Joseph Sylvester by , is a ring in which Sylvester's law of nullity holds. This means that if "A" is an "m" by "n" matrix and "B" an "n" by "s" matrix over "R", then
where ρ is the inner rank of a matrix. The inner rank of an "m" by "n" matrix is the smallest integer "r" such that the matrix is a product of an "m" by "r" matrix and an "r" by "n" matrix.
 showed that fields satisfy Sylvester's law of nullity and are therefore Sylvester domains.