Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.
The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.
It is named after Hermann Weyl.
Definition and examples.
Let formula_1 be a root system in a Euclidean space formula_2. For each root formula_3, let formula_4 denote the reflection about the hyperplane perpendicular to formula_5, which is given explicitly as
where formula_7 is the inner product on formula_2. The Weyl group formula_9 of formula_1 is the subgroup of the orthogonal group formula_11 generated by all the formula_4's. By the definition of a root system, each formula_4 preserves formula_1, from which it follows that formula_9 is a finite group.
In the case of the formula_16 root system, for example, the hyperplanes perpendicular to the roots are just lines, and the Weyl group is the symmetry group of an equilateral triangle, as indicated in the figure. As a group, formula_9 is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle. Note that in this case, formula_9 is not the full symmetry group of the root system; a 60-degree rotation preserves formula_1 but is not an element of formula_9.
We may consider also the formula_21 root system. In this case, formula_2 is the space of all vectors in formula_23 whose entries sum to zero. The roots consist of the vectors of the form formula_24, where formula_25 is the formula_26th standard basis element for formula_23. The reflection associated to such a root is the transformation of formula_2 obtained by interchanging the formula_26th and formula_30th entries of each vector. The Weyl group for formula_21 is then the permutation group on formula_32 elements.
Weyl chambers.
If formula_33 is a root system, we may consider the hyperplane perpendicular to each root formula_5. Recall that formula_4 denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of formula_2 generated by all the formula_4's. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points formula_38 such that formula_39 for all formula_40.
Since the reflections formula_41 preserve formula_1, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.
The figure illustrates the case of the A2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.
A basic general theorem about Weyl chambers is this:
A related result is this one:
Coxeter group structure.
Generating set.
A key result about the Weyl group is this:
That is to say, the group generated by the reflections formula_53 is the same as the group generated by the reflections formula_41.
Relations.
Meanwhile, if formula_5 and formula_56 are in formula_48, then the Dynkin diagram for formula_1 relative to the base formula_48 tells us something about how the pair formula_60 behaves. Specifically, suppose formula_45 and formula_62 are the corresponding vertices in the Dynkin diagram. Then we have the following results:
The preceding claim is not hard to verify, if we simply remember what the Dynkin diagram tells us about the angle between each pair of roots. If, for example, there is no bond between the two vertices, then formula_5 and formula_56 are orthogonal, from which it follows easily that the corresponding reflections commute. More generally, the number of bonds determines the angle formula_81 between the roots. The product of the two reflections is then a rotation by angle formula_82 in the plane spanned by formula_5 and formula_56, as the reader may verify, from which the above claim follows easily.
As a Coxeter group.
Weyl groups are examples of finite reflection groups, as they are generated by reflections; the abstract groups (not considered as subgroups of a linear group) are accordingly finite Coxeter groups, which allows them to be classified by their Coxeter–Dynkin diagram. Being a Coxeter group means that a Weyl group has a special kind of presentation in which each generator "xi" is of order two, and the relations other than "xi2=1" are of the form ("x""i""x""j")"m""ij"=1. The generators are the reflections given by simple roots, and "mij" is 2, 3, 4, or 6 depending on whether roots "i" and "j" make an angle of 90, 120, 135, or 150 degrees, i.e., whether in the Dynkin diagram they are unconnected, connected by a simple edge, connected by a double edge, or connected by a triple edge. We have already noted these relations in the bullet points above, but to say that formula_9 is a Coxeter group, we are saying that those are the "only" relations in formula_9.
Weyl groups have a Bruhat order and length function in terms of this presentation: the "length" of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. There is a unique longest element of a Coxeter group, which is opposite to the identity in the Bruhat order.
Weyl groups in algebraic, group-theoretic, and geometric settings.
Above, the Weyl group was defined as a subgroup of the isometry group of a root system. There are also various definitions of Weyl groups specific to various group-theoretic and geometric contexts (Lie algebra, Lie group, symmetric space, etc.). For each of these ways of defining Weyl groups, it is a (usually nontrivial) theorem that it is a Weyl group in the sense of the definition at the top of this article, namely the Weyl group of some root system associated with the object. A concrete realization of such a Weyl group usually depends on a choice – e.g. of Cartan subalgebra for a Lie algebra, of maximal torus for a Lie group.
The Weyl group of a connected compact Lie group.
Let formula_87 be a connected compact Lie group and let formula_88 be a maximal torus in formula_87. We then introduce the normalizer of formula_88 in formula_87, denoted formula_92 and defined as
We also define the centralizer of formula_88 in formula_87, denoted formula_96 and defined as
The Weyl group formula_9 of formula_87 (relative to the given maximal torus formula_88) is then defined initially as
Eventually, one proves that formula_102, at which point one has an alternative description of the Weyl group as
Now, one can define a root system formula_1 associated to the pair formula_105; the roots are the nonzero weights of the adjoint action of formula_88 on the Lie algebra of formula_87. For each formula_3, one can construct an element formula_109 of formula_92 whose action on formula_88 has the form of reflection. With a bit more effort, one can show that these reflections generate all of formula_112. Thus, in the end, the Weyl group as defined as formula_113 or formula_112 is isomorphic to the Weyl group of the root system formula_1.
In other settings.
For a complex semisimple Lie algebra, the Weyl group is simply "defined" as the reflection group generated by reflections in the roots – the specific realization of the root system depending on a choice of Cartan subalgebra.
For a Lie group "G" satisfying certain conditions, given a torus "T" < "G" (which need not be maximal), the Weyl group "with respect to" that torus is defined as the quotient of the normalizer of the torus "N" = "N"("T") = "NG"("T") by the centralizer of the torus "Z" = "Z"("T") = "ZG"("T"),
The group "W" is finite – "Z" is of finite index in "N". If "T" = "T"0 is a maximal torus (so it equals its own centralizer: formula_117) then the resulting quotient "N"/"Z" = "N"/"T" is called "the" Weyl group of "G", and denoted "W"("G"). Note that the specific quotient set depends on a choice of maximal torus, but the resulting groups are all isomorphic (by an inner automorphism of "G"), since maximal tori are conjugate.
If "G" is compact and connected, and "T" is a "maximal" torus, then the Weyl group of "G" is isomorphic to the Weyl group of its Lie algebra, as discussed above.
For example, for the general linear group "GL," a maximal torus is the subgroup "D" of invertible diagonal matrices, whose normalizer is the generalized permutation matrices (matrices in the form of permutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is the symmetric group. In this case the quotient map "N" → "N"/"T" splits (via the permutation matrices), so the normalizer "N" is a semidirect product of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup of "G". In general this is not always the case – the quotient does not always split, the normalizer "N" is not always the semidirect product of "W" and "Z," and the Weyl group cannot always be realized as a subgroup of "G."
Bruhat decomposition.
If "B" is a Borel subgroup of "G", i.e., a maximal connected solvable subgroup and a maximal torus "T" = "T"0 is chosen to lie in "B", then we obtain the Bruhat decomposition
which gives rise to the decomposition of the flag variety "G"/"B" into Schubert cells (see Grassmannian).
The structure of the Hasse diagram of the group is related geometrically to the cohomology of the manifold (rather, of the real and complex forms of the group), which is constrained by Poincaré duality. Thus algebraic properties of the Weyl group correspond to general topological properties of manifolds. For instance, Poincaré duality gives a pairing between cells in dimension "k" and in dimension "n" - "k" (where "n" is the dimension of a manifold): the bottom (0) dimensional cell corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to the longest element of a Coxeter group.
Analogy with algebraic groups.
There are a number of analogies between algebraic groups and Weyl groups – for instance, the number of elements of the symmetric group is "n"!, and the number of elements of the general linear group over a finite field is related to the "q"-factorial formula_119; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Weyl groups to be simple algebraic groups over the field with one element.
Cohomology.
For a non-abelian connected compact Lie group "G," the first group cohomology of the Weyl group "W" with coefficients in the maximal torus "T" used to define it, is related to the outer automorphism group of the normalizer formula_120 as:
The outer automorphisms of the group Out("G") are essentially the diagram automorphisms of the Dynkin diagram, while the group cohomology is computed in and is a finite elementary abelian 2-group (formula_122); for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.


Masayoshi Nagata

Masayoshi Nagata (Japanese: 永田 雅宜 "Nagata Masayoshi"; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra.
Work.
Nagata's compactification theorem shows that varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a group.
In 1959 he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension.
Nagata's conjecture on curves concerns the minimum degree of a plane curve specified to have given multiplicities at given points; see also Seshadri constant. Nagata's conjecture on automorphisms concerns the existence of wild automorphisms of polynomial algebras in three variables. Recent work has solved this latter problem in the affirmative.


Wilhelm Magnus

Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907, Berlin, Germany – October 15, 1990, New Rochelle, NY) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.
Biography.
In 1931, Magnus received his PhD from the University of Frankfurt, in Germany. His thesis, written under the direction of Max Dehn, was entitled "Über unendlich diskontinuierliche Gruppen von einer definierenden Relation (der Freiheitssatz)".
Magnus was a faculty member in Frankfurt from 1933 until 1938. He refused to join the Nazi Party and, as a consequence, was not allowed to hold an academic post during World War II. In 1947 he became a professor at the University of Göttingen.
In 1948 he emigrated to the United States to collaborate on the Bateman Manuscript Project as a co-editor, while a visiting professor at the California Institute of Technology. In 1950 he was appointed professor at the Courant Institute of Mathematical Sciences, in New York University. He stayed there until 1973, when he moved to the Polytechnic Institute of New York, before retiring in 1978. Among his doctoral students are Joan Birman, Martin Greendlinger, Edna Grossman, Herbert Keller, and Seymour Lipschutz.


Cartan decomposition

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.
Cartan involutions on Lie algebras.
Let formula_1 be a real semisimple Lie algebra and let formula_2 be its Killing form. An involution on formula_1 is a Lie algebra automorphism formula_4 of formula_1 whose square is equal to the identity. Such an involution is called a "Cartan involution" on formula_1 if formula_7 is a positive definite bilinear form.
Two involutions formula_8 and formula_9 are considered equivalent if they differ only by an inner automorphism.
Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.
Cartan pairs.
Let formula_4 be an involution on a Lie algebra formula_1. Since formula_35, the linear map formula_4 has the two eigenvalues formula_37. If formula_38 and formula_39 denote the eigenspaces corresponding to +1 and -1, respectively, then formula_40. Since formula_4 is a Lie algebra automorphism, the Lie bracket of two of its eigenspaces is contained in the eigenspace corresponding to the product of their eigenvalues. It follows that
Thus formula_38 is a Lie subalgebra, while any subalgebra of formula_39 is commutative.
Conversely, a decomposition formula_40 with these extra properties determines an involution formula_4 on formula_1 that is formula_50 on formula_38 and formula_52 on formula_39.
Such a pair formula_54 is also called a "Cartan pair" of formula_1,
and formula_56 is called a "symmetric pair". This notion of a Cartan pair here is not to be confused with the distinct notion involving the relative Lie algebra cohomology formula_57.
The decomposition formula_40 associated to a Cartan involution is called a "Cartan decomposition" of formula_1. The special feature of a Cartan decomposition is that the Killing form is negative definite on formula_38 and positive definite on formula_39. Furthermore, formula_38 and formula_39 are orthogonal complements of each other with respect to the Killing form on formula_1.
Cartan decomposition on the Lie group level.
Let formula_65 be a non-compact semisimple Lie group and formula_1 its Lie algebra. Let formula_4 be a Cartan involution on formula_1 and let formula_69 be the resulting Cartan pair. Let formula_70 be the analytic subgroup of formula_65 with Lie algebra formula_38. Then:
The automorphism formula_73 is also called the "global Cartan involution", and the diffeomorphism formula_79 is called the "global Cartan decomposition". If we write formula_85
this says that the product map formula_86 is a diffeomorphism so formula_87.
For the general linear group, formula_88 is a Cartan involution.
A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras formula_89 in formula_39 are unique up to conjugation by formula_70. Moreover,
where formula_93.
In the compact and noncompact case the global Cartan decomposition thus implies
Geometrically the image of the subgroup formula_95 in formula_96 is a totally geodesic submanifold.
Relation to polar decomposition.
Consider formula_97 with the Cartan involution formula_11. Then formula_99 is the real Lie algebra of skew-symmetric matrices, so that formula_100, while formula_39 is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from formula_39 onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. The polar decomposition of an invertible matrix is unique.


Torsion abelian group

In abstract algebra, a torsion abelian group is an abelian group in which every element has finite order. For example, the torsion subgroup of an abelian group is a torsion abelian group.


Functor

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.
The words "category" and "functor" were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used "functor" in a linguistic context;
see function word.
Definition.
Let "C" and "D" be categories. A functor "F" from "C" to "D" is a mapping that
That is, functors must preserve identity morphisms and composition of morphisms.
Covariance and contravariance.
There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor "F" from "C" to "D" as a mapping that
Note that contravariant functors reverse the direction of composition.
Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a "covariant" functor on the opposite category formula_19. Some authors prefer to write all expressions covariantly. That is, instead of saying formula_20 is a contravariant functor, they simply write formula_21 (or sometimes formula_22) and call it a functor.
Contravariant functors are also occasionally called "cofunctors".
There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections formula_23 of a tangent bundle formula_24—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections formula_25 of a cotangent bundle formula_26—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as formula_27 for formula_28 or formula_29 for formula_30 In this formalism it is observed that the coordinate transformation symbol formula_31 (representing the matrix formula_32) acts on the basis vectors "in the same way" as on the "covector coordinates": formula_33—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: formula_34). This terminology is contrary to the one used in category theory because it is the covectors that have "pullbacks" in general and are thus "contravariant", whereas vectors in general are "covariant" since they can be "pushed forward". See also Covariance and contravariance of vectors.
Opposite functor.
Every functor formula_20 induces the opposite functor formula_36, where formula_19 and formula_38 are the opposite categories to formula_39 and formula_40. By definition, formula_41 maps objects and morphisms identically to formula_42. Since formula_19 does not coincide with formula_39 as a category, and similarly for formula_40, formula_41 is distinguished from formula_42. For example, when composing formula_48 with formula_49, one should use either formula_50 or formula_51. Note that, following the property of opposite category, formula_52.
Bifunctors and multifunctors.
A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type . It can be seen as a functor in "two" arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.
A multifunctor is a generalization of the functor concept to "n" variables. So, for example, a bifunctor is a multifunctor with .
Examples.
Diagram: For categories "C" and "J", a diagram of type "J" in "C" is a covariant functor formula_53.
(Category theoretical) presheaf: For categories "C" and "J", a "J"-presheaf on "C" is a contravariant functor formula_54.
Presheaves: If "X" is a topological space, then the open sets in "X" form a partially ordered set Open("X") under inclusion. Like every partially ordered set, Open("X") forms a small category by adding a single arrow if and only if formula_55. Contravariant functors on Open("X") are called "presheaves" on "X". For instance, by assigning to every open set "U" the associative algebra of real-valued continuous functions on "U", one obtains a presheaf of algebras on "X".
Constant functor: The functor which maps every object of "C" to a fixed object "X" in "D" and every morphism in "C" to the identity morphism on "X". Such a functor is called a "constant" or "selection" functor.
Endofunctor: A functor that maps a category to that same category; e.g., polynomial functor.
Identity functor: in category "C", written 1"C" or id"C", maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
Diagonal functor: The diagonal functor is defined as the functor from "D" to the functor category "D""C" which sends each object in "D" to the constant functor at that object.
Limit functor: For a fixed index category "J", if every functor has a limit (for instance if "C" is complete), then the limit functor assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitable version of the axiom of choice. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant).
Power sets functor: The power set functor maps each set to its power set and each function formula_56 to the map which sends formula_57 to its image formula_58. One can also consider the contravariant power set functor which sends formula_59 to the map which
sends formula_60 to its inverse image formula_61
For example, if formula_62 then formula_63. Suppose formula_64 and formula_65. Then formula_66 is the function which sends any subset formula_67 of formula_1 to its image formula_69, which in this case means
formula_70, where formula_71 denotes the mapping under formula_66, so this could also be written as formula_73. For the other values, formula_74 Note that formula_75 consequently generates the trivial topology on formula_1. Also note that although the function formula_77 in this example mapped to the power set of formula_1, that need not be the case in general.
Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs , where "X" is a topological space and "x"0 is a point in "X". A morphism from to is given by a continuous map with .
To every topological space "X" with distinguished point "x"0, one can define the fundamental group based at "x"0, denoted . This is the group of homotopy classes of loops based at "x"0, with the group operation of concatenation. If is a morphism of pointed spaces, then every loop in "X" with base point "x"0 can be composed with "f" to yield a loop in "Y" with base point "y"0. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from to . We thus obtain a functor from the category of pointed topological spaces to the category of groups.
In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial.
Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space "X" the algebra C("X") of all real-valued continuous functions on that space. Every continuous map induces an algebra homomorphism by the rule for every "φ" in C("Y").
Tangent and cotangent bundles: The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles.
Doing this constructions pointwise gives the tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above.
Group actions/representations: Every group "G" can be considered as a category with a single object whose morphisms are the elements of "G". A functor from "G" to Set is then nothing but a group action of "G" on a particular set, i.e. a "G"-set. Likewise, a functor from "G" to the category of vector spaces, Vect"K", is a linear representation of "G". In general, a functor can be considered as an "action" of "G" on an object in the category "C". If "C" is a group, then this action is a group homomorphism.
Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor.
Tensor products: If "C" denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product formula_79 defines a functor which is covariant in both arguments.
Forgetful functors: The functor which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed "forgetful functors". Another example is the functor which maps a ring to its underlying additive abelian group. Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).
Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor sends every set "X" to the free group generated by "X". Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object.
Homomorphism groups: To every pair "A", "B" of abelian groups one can assign the abelian group Hom("A", "B") consisting of all group homomorphisms from "A" to "B". This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor (where Ab denotes the category of abelian groups with group homomorphisms). If and are morphisms in Ab, then the group homomorphism : is given by . See Hom functor.
Representable functors: We can generalize the previous example to any category "C". To every pair "X", "Y" of objects in "C" one can assign the set of morphisms from "X" to "Y". This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor . If and are morphisms in "C", then the map is given by .
Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable.
Properties.
Two important consequences of the functor axioms are:
One can compose functors, i.e. if "F" is a functor from "A" to "B" and "G" is a functor from "B" to "C" then one can form the composite functor from "A" to "C". Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories.
A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
Relation to other categorical concepts.
Let "C" and "D" be categories. The collection of all functors from "C" to "D" forms the objects of a category: the functor category. Morphisms in this category are natural transformations between functors.
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above.
Universal constructions often give rise to pairs of adjoint functors.
Computer implementations.
Functors sometimes appear in functional programming. For instance, the programming language Haskell has a class codice_1 where codice_2 is a polytypic function used to map functions ("morphisms" on "Hask", the category of Haskell types) between existing types to functions between some new types.


Teruhisa Matsusaka

 (1926–2006) was a Japanese-born American mathematician, who specialized in algebraic geometry.
Matsusaka was awarded a Guggenheim Fellowship for the academic year 1959–1960.
In 1972, Matsusaka introduced Matsusaka's big theorem, a key technical result on ample line bundles.


Richard Lyons (mathematician)

Richard Neil Lyons (born January 22, 1945 in New York City, New York) is an American mathematician, specializing in finite group theory.
Lyons received his PhD in 1970 at the University of Chicago under John Griggs Thompson with a thesis entitled "Characterizations of Some Finite Simple Groups with Small 2-Rank". From 1972 to 2017 he was a professor at Rutgers University.
With Daniel Gorenstein and Ronald Solomon he wrote, and is continuing to write, a series on the classification program for finite simple groups, a program in which the three of them were major participants. Nine volumes of this series have been published so far. He discovered a sporadic group which Charles Sims constructed and called the Lyons group "Ly".
In 2012 he shared the Leroy P. Steele Prize for Mathematical Exposition, awarded by the American Mathematical Society, with Michael Aschbacher, Stephen D. Smith, and Ronald Solomon.
In 2013 he became a fellow of the American Mathematical Society "for contributions to the classification of the finite simple groups, including the discovery of one of the 26 sporadic finite simple groups.".


Commuting matrices

In linear algebra, two matrices formula_1 and formula_2 are said to commute if formula_3, or equivalently if their commutator formula_4 is zero. A set of matrices formula_5 is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.
History.
The notion of commuting matrices was introduced by Cayley in his memoir on the theory of matrices, which also provided the first axiomatization of matrices. The first significant results proved on them was the above result of Frobenius in 1878.


Dowling geometry

In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines the group. Dowling geometries have a role in matroid theory as universal objects (Kahn and Kung, 1982); in that respect they are analogous to projective geometries, but based on groups instead of fields.
A Dowling lattice is the geometric lattice of flats associated with a Dowling geometry. The lattice and the geometry are mathematically equivalent: knowing either one determines the other. Dowling lattices, and by implication Dowling geometries, were introduced by Dowling (1973a,b).
A Dowling lattice or geometry of rank "n" of a group "G" is often denoted "Qn"("G").
The original definitions.
In his first paper (1973a) Dowling defined the rank-"n" Dowling lattice of the multiplicative group of a finite field "F". It is the set of all those subspaces of the vector space "F""n" that are generated by subsets of the set "E" that consists of vectors with at most two nonzero coordinates. The corresponding Dowling geometry is the set of 1-dimensional vector subspaces generated by the elements of "E".
In his second paper (1973b) Dowling gave an intrinsic definition of the rank-"n" Dowling lattice of any finite group "G". Let "S" be the set {1...,"n"}. A "G"-labelled set ("T", "α") is a set "T" together with a function "α": "T" → "G". Two "G"-labelled sets, ("T", "α") and ("T", "β"), are equivalent if there is a group element, "g", such that "β" = "gα".
An equivalence class is denoted ["T", "α"].
A partial "G"-partition of "S" is a set "γ" = {["B"1,"α"1], ..., ["B""k","α""k"]} of equivalence classes of "G"-labelled sets such that "B"1, ..., "B""k" are nonempty subsets of "S" that are pairwise disjoint. ("k" may equal 0.)
A partial "G"-partition "γ" is said to be ≤ another one, "γ"*, if
This gives a partial ordering of the set of all partial "G"-partitions of "S". The resulting partially ordered set is the Dowling lattice "Q""n"("G").
The definitions are valid even if "F" or "G" is infinite, though Dowling mentioned only finite fields and groups.
Graphical definitions.
A graphical definition was then given by Doubilet, Rota, and Stanley (1972). We give the slightly simpler (but essentially equivalent) graphical definition of Zaslavsky (1991), expressed in terms of gain graphs.
Take "n" vertices, and between each pair of vertices, "v" and "w", take a set of |"G"| parallel edges labelled by each of the elements of the group "G". The edges are oriented, in that, if the label in the direction from "v" to "w" is the group element "g", then the label of the same edge in the opposite direction, from "w" to "v", is "g"−1. The label of an edge therefore depends on the direction of the edge; such labels are called gains. Also add to each vertex a loop whose gain is any value other than 1. (1 is the group identity element.) This gives a graph which is called "GKn"o (note the raised circle).
A cycle in the graph then has a gain. The cycle is a sequence of edges, "e"1"e"2···"e""k". Suppose the gains of these edges, in a fixed direction around the cycle, are "g"1, "g"2, ..., "g""k". Then the gain of the cycle is the product, "g"1"g"2···"g""k". The value of this gain is not completely well defined, since it depends on the direction chosen for the cycle and on which is called the "first" edge of the cycle. What is independent of these choices is the answer to the following question: is the gain equal to 1 or not? If it equals 1 under one set of choices, then it is also equal to 1 under all sets of choices.
To define the Dowling geometry, we specify the circuits (minimal dependent sets). The circuits of the matroid are
Thus, the Dowling geometry "Qn"("G") is the frame matroid (or bias matroid) of the gain graph "GKn"o (the raised circle denotes the presence of loops).
Other, equivalent definitions are described in the article on gain graphs.
Characteristic polynomial.
One reason for interest in Dowling lattices is that the characteristic polynomial is very simple. If "L" is the Dowling lattice of rank "n" of a finite group "G" having "m" elements, then
an exceptionally simple formula for any geometric lattice.
Generalizations.
There is also a Dowling geometry, of rank 3 only, associated with each quasigroup; see Dowling (1973b). This does not generalize in a straightforward way to higher ranks. There is a generalization due to Zaslavsky (2012) that involves "n"-ary quasigroups.


Multiply transitive group


Complete variety

In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety "X", such that for any variety "Y" the projection morphism
is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space "X" is compact if and only if the above projection map is closed with respect to topological products.
The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete.
A complex variety is complete if and only if it is compact as a complex-analytic variety.
The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. The first examples of non-projective complete varieties were given by Masayoshi Nagata and Heisuke Hironaka. An affine space of positive dimension is not complete.
The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of "complete", in the sense of "no missing points", can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley.


*-regular semigroup


Direct sum of groups

In mathematics, a group "G" is called the direct sum of two subgroups "H"1 and "H"2 if
More generally, "G" is called the direct sum of a finite set of subgroups {"H""i"} if
If "G" is the direct sum of subgroups "H" and "K" then we write , and if "G" is the direct sum of a set of subgroups {"H""i"} then we often write "G" = ∑"H""i". Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.
In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
This direct sum is commutative up to isomorphism. That is, if then also and thus . It is also associative in the sense that if , and , then .
A group which can be expressed as a direct sum of non-trivial subgroups is called "decomposable", and if a group cannot be expressed as such a direct sum then it is called "indecomposable".
If , then it can be proven that:
The above assertions can be generalized to the case of , where {"H"i} is a finite set of subgroups:
Note the similarity with the direct product, where each "g" can be expressed uniquely as
Since for all , it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product; thus for finite sets of subgroups, ∑"H""i" is isomorphic to the direct product ×{"H""i"}.
Direct summand.
Given a group formula_2, we say that a subgroup formula_3 is a direct summand of formula_2 if there exists another subgroup formula_5 of formula_2 such that formula_7.
In abelian groups, if formula_3 is a divisible subgroup of formula_2, then formula_3 is a direct summand of formula_2.
Equivalence of decompositions into direct sums.
In the decomposition of a finite group into a direct sum of indecomposable subgroups the embedding of the subgroups is not unique. For example, in the Klein group formula_25 we have that
However, the Remak-Krull-Schmidt theorem states that given a "finite" group "G" = ∑"A""i" = ∑"B""j", where each "A""i" and each "B""j" is non-trivial and indecomposable, the two sums have equal terms up to reordering and isomorphism.
The Remak-Krull-Schmidt theorem fails for infinite groups; so in the case of infinite "G" = "H" + "K" = "L" + "M", even when all subgroups are non-trivial and indecomposable, we cannot conclude that "H" is isomorphic to either "L" or "M".
Generalization to sums over infinite sets.
To describe the above properties in the case where "G" is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed.
If "g" is an element of the cartesian product ∏{"H""i"} of a set of groups, let "g""i" be the "i"th element of "g" in the product. The external direct sum of a set of groups {"H""i"} (written as ∑E{"H""i"}) is the subset of ∏{"H""i"}, where, for each element "g" of ∑E{"H""i"}, "g""i" is the identity formula_28 for all but a finite number of "g""i" (equivalently, only a finite number of "g""i" are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
This subset does indeed form a group, and for a finite set of groups {"H""i"} the external direct sum is equal to the direct product.
If "G" = ∑"H""i", then "G" is isomorphic to ∑E{"H""i"}. Thus, in a sense, the direct sum is an "internal" external direct sum. For each element "g" in "G", there is a unique finite set "S" and a unique set {"h""i" ∈ "H""i" : "i" ∈ "S"} such that "g" = ∏ {"h""i" : "i" in "S"}.


Sorin Popa

Sorin Teodor Popa (24 March 1953) is a Romanian American mathematician working on operator algebras. He is a professor at the University of California, Los Angeles.
Biography.
Popa earned his Ph.D. from the University of Bucharest in 1983 under the supervision of Dan-Virgil Voiculescu. He has advised 15 doctoral students at UCLA, including Adrian Ioana.
Honors and awards.
In 1990 Popa was an invited speaker at the International Congress of Mathematicians (ICM) in Kyoto, where he gave a talk on "Subfactors and Classifications in von Neumann algebras". He was a Guggenheim Fellow in 1995. In 2006 he gave a plenary lecture at the ICM in Madrid on "Deformation and Rigidity for group actions and Von Neumann Algebras". In 2009 he was awarded the Ostrowski Prize, and in 2010 the E. H. Moore Prize. He is one of the inaugural fellows of the American Mathematical Society.
In 2011 Popa was awarded the National Order of Faithful Service, Knight rank by the President of Romania, Traian Băsescu.


Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set "S" to itself.
In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ).
Automorphisms.
An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication:
Endomorphism rings.
Any two endomorphisms of an abelian group, , can be added together by the rule . Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of is the ring of all matrices with integer entries. The endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a near-ring. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group; however there are rings that are not the endomorphism ring of any abelian group.
Operator theory.
In any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits of elements, etc.
Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory.
Endofunctions.
An endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism.
Let be an arbitrary set. Among endofunctions on one finds permutations of and constant functions associating to every in the same element in . Every permutation of has the codomain equal to its domain and is bijective and invertible. If has more than one element, a constant function on has an image that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each natural number the floor of has its image equal to its codomain and is not invertible.
Finite endofunctions are equivalent to directed pseudoforests. For sets of size there are endofunctions on the set.
Particular examples of bijective endofunctions are the involutions; i.e., the functions coinciding with their inverses.


Complete graph

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).
Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose.
Properties.
The complete graph on formula_1 vertices is denoted by formula_2. Some sources claim that the letter formula_3 in this notation stands for the German word "komplett", but the German name for a complete graph, "vollständiger Graph", does not contain the letter formula_3, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory.
formula_5 has formula_6 edges (a triangular number), and is a regular graph of degree formula_7. All complete graphs are their own maximal cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph.
If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament.
formula_5 can be decomposed into formula_1 trees formula_10 such that formula_10 has formula_12 vertices. Ringel's conjecture asks if the complete graph formula_13 can be decomposed into copies of any tree with formula_1 edges. This is known to be true for sufficiently large formula_1.
The number of matchings of the complete graphs are given by the telephone numbers
These numbers give the largest possible value of the Hosoya index for an formula_1-vertex graph. The number of perfect matchings of the complete graph formula_5 (with formula_1 even) is given by the double factorial formula_19.
The crossing numbers up to formula_20 are known, with formula_21 requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project. Rectilinear Crossing numbers for formula_5 are
Geometry and topology.
A complete graph with formula_1 nodes represents the edges of an formula_24-simplex. Geometrically forms the edge set of a triangle, a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton.
 through are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither nor the complete bipartite graph as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions. As part of the Petersen family, plays a similar role as one of the forbidden minors for linkless embedding. In other words, and as Conway and Gordon proved, every embedding of into three-dimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any three-dimensional embedding of contains a Hamiltonian cycle that is embedded in space as a nontrivial knot.
Examples.
Complete graphs on formula_1 vertices, for formula_1 between 1 and 12, are shown below along with the numbers of edges:


Jakob Stix

Jakob M. Stix (born in 1974) is a German mathematician. He deals with arithmetic algebraic geometry (étale fundamental group, anabelian geometry and others).
Stix studied mathematics in Freiburg and Bonn and received his doctorate in 2002 from Florian Pop at the University of Bonn ("Projective Anabelian Curves in Positive Characteristic and Descent Theory for Log-Etale Covers"). His dissertation was awarded the best doctoral thesis of the year 2002 by the Mathematical Institute of the University of Bonn. He was a post-doctoral student at the Institute for Advanced Study. In 2008 he became junior research group leader at the Mathematics Center of the University of Heidelberg, where he habilitated in 2011 ("Evidence for the section conjecture in the theory of arithmetic fundamental groups"). Stix is now a professor at the University of Frankfurt.
He recently discovered and contributed to the solution of an error in an influential 1997 paper by Lucia Caporaso, Joe Harris and Barry Mazur on the implications of the Strong Lang Conjecture ("Uniformity of rational points". J. Amer. Math. Soc.,101-5 (1997))


Cyrus Colton MacDuffee

Cyrus Colton MacDuffee (June 29, 1895 – August 21, 1961) from Oneida, New York was a professor of mathematics at University of Wisconsin.
He wrote a number of influential research papers in abstract algebra. MacDuffee served on the Council of the American Mathematical Society (A.M.S.), was editor of the "Transactions of the A.M.S.", and served as president of the Mathematical Association of America (M.A.A).
MacDuffee obtained his B.S. degree in 1917 from Colgate University and a Ph.D. in 1922 from the University of Chicago ; his thesis was on Nonassociative algebras under the direction of Leonard E. Dickson. In 1935, MacDuffee joined the University of Wisconsin, where he remained until his death in 1961. He served as chair of the department (1951–56). Later, Wisconsin endowed a university chair under his name. Prior to joining the University of Wisconsin, he served at Princeton and Ohio State. He guided 30 Ph.D. students, among them D. R. Fulkerson, H. J. Ryser, and Bonnie Stewart.
MacDuffee's daughter Helen became a statistician at Oregon State University and the mayor of Corvallis, Oregon.


Theorem of absolute purity

In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given
for each integer formula_4, the map
is bijective, where the map is induced by cup product with formula_6.
The theorem was introduced in SGA 5 Exposé I, § 3.1.4. as an open problem. Later, Thomason proved it for large "n" and Gabber in general.


Character theory

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.
Applications.
Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow -subgroup.
Definitions.
Let be a finite-dimensional vector space over a field and let be a representation of a group on . The character of is the function given by
where is the trace.
A character is called irreducible or simple if is an irreducible representation. The degree of the character is the dimension of ; in characteristic zero
this is equal to the value . A character of degree 1 is called linear. When is finite and has characteristic zero, the kernel of the character is the normal subgroup:
which is precisely the kernel of the representation . However, the character is "not" a group homomorphism in general.
Properties.
Arithmetic properties.
Let ρ and σ be representations of . Then the following identities hold:


General semilinear group


Grassmann bundle

In algebraic geometry, the Grassmann "d"-plane bundle of a vector bundle "E" on an algebraic scheme "X" is a scheme over "X":
such that the fiber formula_2 is the Grassmannian of the "d"-dimensional vector subspaces of formula_3. For example, formula_4 is the projective bundle of "E". In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.
Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle "S" and universal quotient bundle "Q" that fit into
Specifically, if "V" is in the fiber "p"−1("x"), then the fiber of "S" over "V" is "V" itself; thus, "S" has rank "r" = rk("E") and formula_6 is the determinant line bundle. Now, by the universal property of a projective bundle, the injection formula_7 corresponds to the morphism over "X":
which is nothing but a family of Plücker embeddings.
The relative tangent bundle "T""G""d"("E")/"X" of "G""d"("E") is given by
which morally is given by the second fundamental form. In the case "d" = 1, it is given as follows: if "V" is a finite-dimensional vector space, then for each line formula_10 in "V" passing through the origin (a point of formula_11), there is the natural identification (see Chern class#Complex projective space for example):
and the above is the family-version of this identification. (The general care is a generalization of this.)
In the case "d" = 1, the early exact sequence tensored with the dual of "S" = "O"(-1) gives:
which is the relative version of the Euler sequence.


Goldman domain

In mathematics, a Goldman domain or G-domain is an integral domain "A" whose field of fractions is a finitely generated algebra over "A". They are named after Oscar Goldman.
An overring (i.e., an intermediate ring lying between the ring and its field of fractions) of a Goldman domain is again a Goldman domain. There exists a Goldman domain where all nonzero prime ideals are maximal although there are infinitely many prime ideals.
An ideal "I" in a commutative ring "A" is called a Goldman ideal if the quotient "A"/"I" is a Goldman domain. A Goldman ideal is thus prime, but not necessarily maximal. In fact, a commutative ring is a Jacobson ring if and only if every Goldman ideal in it is maximal.
The notion of a Goldman ideal can be used to give a slightly sharpened characterization of a radical of an ideal: the radical of an ideal "I" is the intersection of all Goldman ideals containing "I".
Alternative definition.
An integral domain formula_1 is a G-domain if and only if:
A G-ideal is defined as an ideal formula_8 such that formula_9 is a G-domain. Since a factor ring is an integral domain if and only if the ring is factored by a prime ideal, every G-ideal is also a prime ideal. G-ideals can be used as a refined collection of prime ideals in the following sense: Radical can be characterized as the intersection of all prime ideals containing the ideal, and in fact we still get the radical even if we take the intersection over the G-ideals.
Every maximal ideal is a G-ideal, since quotient by maximal ideal is a field, and a field is trivially a G-domain. Therefore, maximal ideals are G-ideals, and G-ideals are prime ideals. G-ideals are the only maximal ideals in Jacobson ring, and in fact this is an equivalent characterization of a Jacobson ring: a ring is a Jacobson ring when all G-ideals are maximal ideals. This leads to a simplified proof of the Nullstellensatz.
It is known that given formula_10, a ring extension of a G-domain, formula_11 is algebraic over formula_12 if and only if every ring extension between formula_11 and formula_12 is a G-domain.
A Noetherian domain is a G-domain iff its rank is at most one, and has only finitely many maximal ideals (or equivalently, prime ideals).


Reinhold Baer

Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups.
Biography.
Baer studied mechanical engineering for a year at Leibniz University Hannover. He then went to study philosophy at Freiburg in 1921. While he was at Göttingen in 1922 he was influenced by Emmy Noether and Hellmuth Kneser. In 1924 he won a scholarship for specially gifted students. Baer wrote up his doctoral dissertation and it was published in Crelle's Journal in 1927.
Baer accepted a post at Halle in 1928. There, he published Ernst Steinitz's "Algebraische Theorie der Körper" with Helmut Hasse, first published in Crelle's Journal in 1910.
While Baer was with his wife in Austria, Adolf Hitler and the Nazis came into power. Both of Baer's parents were Jewish, and he was for this reason informed that his services at Halle were no longer required. Louis Mordell invited him to go to Manchester and Baer accepted.
Baer stayed at Princeton University and was a visiting scholar at the nearby Institute for Advanced Study from 1935 to 1937. For a short while he lived in North Carolina. From 1938 to 1956 he worked at the University of Illinois at Urbana-Champaign. He returned to Germany in 1956.
According to biographer K. W. Gruenberg,
He died of heart failure on October 22nd in 1979.
In 2016 the Reinhold Baer Prize for the best Ph.D. thesis in Group Theory was set up in his honour.


Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, "see Relative concreteness below"). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets.
A concrete category, when defined without reference to the notion of a category, consists of a class of "objects", each equipped with an "underlying set"; and for any two objects "A" and "B" a set of functions, called "morphisms", from the underlying set of "A" to the underlying set of "B". Furthermore, for every object "A", the identity function on the underlying set of "A" must be a morphism from "A" to "A", and the composition of a morphism from "A" to "B" followed by a morphism from "B" to "C" must be a morphism from "A" to "C".
Definition.
A concrete category is a pair ("C","U") such that
The functor "U" is to be thought of as a forgetful functor, which assigns to every object of "C" its "underlying set", and to every morphism in "C" its "underlying function".
A category "C" is concretizable if there exists a concrete category ("C","U");
i.e., if there exists a faithful functor "U": "C" → Set. All small categories are concretizable: define "U" so that its object part maps each object "b" of "C" to the set of all morphisms of "C" whose codomain is "b" (i.e. all morphisms of the form "f": "a" → "b" for any object "a" of "C"), and its morphism part maps each morphism "g": "b" → "c" of "C" to the function "U"("g"): "U"("b") → "U"("c") which maps each member "f": "a" → "b" of "U"("b") to the composition "gf": "a" → "c", a member of "U"("c"). (Item 6 under Further examples expresses the same "U" in less elementary language via presheaves.) The Counter-examples section exhibits two large categories that are not concretizable.
Remarks.
It is important to note that, contrary to intuition, concreteness is not a property which a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. In particular, a category "C" may admit several faithful functors into Set. Hence there may be several concrete categories ("C", "U") all corresponding to the same category "C".
In practice, however, the choice of faithful functor is often clear and in this case we simply speak of the "concrete category "C"". For example, "the concrete category Set" means the pair (Set, "I") where "I" denotes the identity functor Set → Set.
The requirement that "U" be faithful means that it maps different morphisms between the same objects to different functions. However, "U" may map different objects to the same set and, if this occurs, it will also map different morphisms to the same function.
For example, if "S" and "T" are two different topologies on the same set "X", then
("X", "S") and ("X", "T") are distinct objects in the category Top of topological spaces and continuous maps, but mapped to the same set "X" by the forgetful functor Top → Set. Moreover, the identity morphism ("X", "S") → ("X", "S") and the identity morphism ("X", "T") → ("X", "T") are considered distinct morphisms in Top, but they have the same underlying function, namely the identity function on "X".
Similarly, any set with four elements can be given two non-isomorphic group structures: one isomorphic to formula_1, and the other isomorphic to formula_2.
Counter-examples.
The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable.
While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions.
The fact that there does not exist "any" faithful functor from hTop to Set was first proven by Peter Freyd.
In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable.
Implicit structure of concrete categories.
Given a concrete category ("C", "U") and a cardinal number "N", let "UN" be the functor "C" → Set determined by "UN(c) = (U(c))N".
Then a subfunctor of "UN" is called an "N-ary predicate" and a
natural transformation "UN" → "U" an "N-ary operation".
The class of all "N"-ary predicates and "N"-ary operations of a concrete category ("C","U"), with "N" ranging over the class of all cardinal numbers, forms a large signature. The category of models for this signature then contains a full subcategory which is equivalent to "C".
Relative concreteness.
In some parts of category theory, most notably topos theory, it is common to replace the category Set with a different category "X", often called a "base category".
For this reason, it makes sense to call a pair ("C", "U") where "C" is a category and "U" a faithful functor "C" → "X" a concrete category over "X".
For example, it may be useful to think of the models of a theory with "N" sorts as forming a concrete category over Set"N".
In this context, a concrete category over Set is sometimes called a "construct".


Hurwitz matrix

In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Hurwitz matrix and the Hurwitz stability criterion.
Namely, given a real polynomial
the formula_2 square matrix
is called Hurwitz matrix corresponding to the polynomial formula_4. It was established by Adolf Hurwitz in 1895 that a real polynomial with formula_5 is stable
(that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix formula_6 are positive:
and so on. The minors formula_8 are called the Hurwitz determinants. Similarly, if formula_9 then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.
Hurwitz stable matrices.
In engineering and stability theory, a square matrix formula_10 is called a stable matrix (or sometimes a Hurwitz matrix) if every eigenvalue of formula_10 has strictly negative real part, that is,
for each eigenvalue formula_13. formula_10 is also called a stability matrix, because then the differential equation
is asymptotically stable, that is, formula_16 as formula_17
If formula_18 is a (matrix-valued) transfer function, then formula_19 is called Hurwitz if the poles of all elements of formula_19 have negative real part. Note that it is not necessary that formula_21 for a specific argument formula_22 be a Hurwitz matrix — it need not even be square. The connection is that if formula_10 is a Hurwitz matrix, then the dynamical system
has a Hurwitz transfer function.
Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point.
The Hurwitz stability matrix is a crucial part of control theory. A system is "stable" if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently "unstable" if any of the eigenvalues have positive real components, representing positive feedback.


Seki Takakazu

, also known as , was a Japanese mathematician and author of the Edo period.
Seki laid foundations for the subsequent development of Japanese mathematics, known as "wasan". He has been described as "Japan's Newton".
He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. Although he was a contemporary of German polymath mathematician and philosopher Gottfried Leibniz and British mathematician Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo period.
While it is not clear how much of the achievements of "wasan" are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered in Europe. For example, he is credited with the discovery of Bernoulli numbers. The resultant and determinant (the first in 1683, the complete version no later than 1710) are attributed to him.
Biography.
Not much is known about Seki's personal life. His birthplace has been indicated as either Fujioka in Gunma Prefecture, or Edo. His birth date ranges from 1635 to 1643.
He was born to the Uchiyama clan, a subject of Ko-shu "han", and adopted into the Seki family, a subject of the "shōgun". While in Ko-shu "han", he was involved in a surveying project to produce a reliable map of his employer's land. He spent many years in studying 13th-century Chinese calendars to replace the less accurate one used in Japan at that time.
Career.
Chinese mathematical roots.
His mathematics (and "wasan" as a whole) was based on mathematical knowledge accumulated from the 13th to 15th centuries. The material in these works consisted of algebra with numerical methods, polynomial interpolation and its applications, and indeterminate integer equations. Seki's work is more or less based on and related to these known methods.
Chinese algebraists discovered numerical evaluation (Horner's method, re-established by William George Horner in the 19th century) of arbitrary-degree algebraic equation with real coefficients. By using the Pythagorean theorem, they reduced geometric problems to algebra systematically. The number of unknowns in an equation was, however, quite limited. They used notations of an array of numbers to represent a formula; for example, formula_1 for formula_2.
Later, they developed a method that uses two-dimensional arrays, representing four variables at most, but the scope of this method was limited. Accordingly, a target of Seki and his contemporary Japanese mathematicians was the development of general multivariable algebraic equations and elimination theory.
In the Chinese approach to polynomial interpolation, the motivation was to predict the motion of celestial bodies from observed data. The method was also applied to find various mathematical formulas. Seki learned this technique, most likely, through his close examination of Chinese calendars.
Competing with contemporaries.
In 1671, , a pupil of in Osaka, published "Kokon Sanpō Ki"　(古今算法記), in which he gave the first comprehensive account of Chinese algebra in Japan. He successfully applied it to problems suggested by his contemporaries. Before him, these problems were solved using arithmetical methods. In the end of the book, he challenged other mathematicians with 15 new problems, which require multi-variable algebraic equations.
In 1674, Seki published "Hatsubi Sanpō" (発微算法), giving solutions to all the 15 problems. The method he used is called "bōsho-hō". He introduced the use of "kanji" to represent unknowns and variables in equations. Although it was possible to represent equations of an arbitrary degree (he once treated the 1458th degree) with negative coefficients, there were no symbols corresponding to parentheses, equality, or division. For example, formula_3 could also mean formula_4. Later, the system was improved by other mathematicians, and in the end it became as expressive as the ones developed in Europe.
In his book of 1674, however, Seki gave only single-variable equations resulting from elimination, but no account of the process at all, nor his new system of algebraic symbols. There were a few errors in the first edition. A mathematician in Hashimoto's school criticized the work, saying "only three out of 15 are correct." In 1678, , who was from Hashimoto's school and was active in Kyoto, authored "Sanpō Meikai" (算法明記), and gave new solutions to Sawaguchi's 15 problems, using his version of multivariable algebra, similar to Seki's. To answer criticism, in 1685, , one of Seki's pupils, published "Hatsubi Sanpō Genkai" (発微算法諺解), notes on "Hatsubi Sanpō", in which he showed in detail the process of elimination using algebraic symbols.
The effect of the introduction of the new symbolism was not restricted to algebra. With it, mathematicians at that time became able to express mathematical results in more general and abstract way. They concentrated on the study of elimination of variables.
Elimination theory.
In 1683, Seki pushed ahead with elimination theory, based on resultants, in the "Kaifukudai no Hō" (解伏題之法). To express the resultant, he developed the notion of the determinant. While in his manuscript the formula for 5×5 matrices is obviously wrong, being always 0, in his later publication, "Taisei Sankei" (大成算経), written in 1683-1710 with Katahiro Takebe (建部 賢弘) and his brothers, a correct and general formula (Laplace's formula for the determinant) appears.
Tanaka came up with the same idea independently. An indication appeared in his book of 1678: some of equations after elimination are the same as resultant. In "Sanpō Funkai" (算法紛解) (1690?), he explicitly described the resultant and applied it to several problems. In 1690, , a mathematician active in Osaka but not in Hashimoto's school, published "Sanpō Hakki" (算法発揮), in which he gave resultant and Laplace's formula of determinant for the "n"×"n" case. The relationships between these works are not clear. Seki developed his mathematics in competition with mathematicians in Osaka and Kyoto, at the cultural center of Japan.
In comparison with European mathematics, Seki's first manuscript was as early as Leibniz's first commentary on the subject, which treated matrices only up to the 3x3 case. The subject was forgotten in the West until Gabriel Cramer in 1750 was brought to it by the same motivations. Elimination theory equivalent to the "wasan" form was rediscovered by Étienne Bézout in 1764. The Laplace's formula was established not earlier than 1750.
With elimination theory in hand, a large part of the problems treated in Seki's time became solvable in principle, given the Chinese tradition of geometry almost reduced to algebra. In practice, the method could founder under huge computational complexity. Yet this theory had a significant influence on the direction of development of "wasan". After the elimination is complete, one is left to find numerically the real roots of a single-variable equation. Horner's method, though well known in China, was not transmitted to Japan in its final form. So Seki had to work it out by himself independently. He is sometimes credited with Horner's method, which is not historically correct. He also suggested an improvement to Horner's method: to omit higher order terms after some iterations. This practice happens to be the same as that of Newton-Raphson method, but with a completely different perspective. Neither he nor his pupils had, strictly speaking, the idea of derivative.
Seki also studied the properties of algebraic equations for assisting in numerical solution. The most notable of these are the conditions for the existence of multiple roots based on the discriminant, which is the resultant of a polynomial and its "derivative": His working definition of "derivative" was the "O(h)" -term in "f"("x" + "h"), which was computed by the binomial theorem.
He obtained some evaluations of the number of real roots of a polynomial equation.
Calculation of pi.
Another of Seki's contributions was the rectification of the circle, i.e., the calculation of pi; he obtained a value for π that was correct to the 10th decimal place, using what is now called the Aitken's delta-squared process, rediscovered in the 20th century by Alexander Aitken.
Legacy.
The asteroid 7483 Sekitakakazu is named after Seki Takakazu.
Selected works.
In a statistical overview derived from writings by and about Seki Takakazu, OCLC/WorldCat encompasses roughly 50+ works in 50+ publications in three languages and 100+ library holdings.


Cyclic subspace

In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector "v" in a vector space "V" and a linear transformation "T" of "V" is called the "T"-cyclic subspace generated by "v". The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.
Definition.
Let formula_1 be a linear transformation of a vector space formula_2 and let formula_3 be a vector in formula_2. The formula_5-cyclic subspace of formula_2 generated by formula_7 is the subspace formula_8 of formula_2 generated by the set of vectors formula_10. This subspace is denoted by formula_11. In the case when formula_2 is a topological vector space, formula_7 is called a cyclic vector for formula_5 if formula_15 is dense in formula_2. For the particular case of finite-dimensional spaces, this is equivalent to saying that formula_15 is the whole space formula_2.
There is another equivalent definition of cyclic spaces. Let formula_1 be a linear transformation of a topological vector space over a field formula_20 and formula_7 be a vector in formula_2. The set of all vectors of the form formula_23, where formula_24 is a polynomial in the ring formula_25 of all polynomials in formula_26 over formula_20, is the formula_5-cyclic subspace generated by formula_7.
The subspace formula_15 is an invariant subspace for formula_5, in the sense that formula_32.
Companion matrix.
Let formula_54 be a linear transformation of a formula_55-dimensional vector space formula_2 over a field formula_20 and formula_7 be a cyclic vector for formula_5. Then the vectors
form an ordered basis for formula_2. Let the characteristic polynomial for formula_5 be
Then
Therefore, relative to the ordered basis formula_65, the operator formula_5 is represented by the matrix
This matrix is called the "companion matrix" of the polynomial formula_68.


Quasiregular element

In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. Intuitively, quasiregularity captures what it means for an element of a ring to be "bad"; that is, have undesirable properties. Although a "bad element" is necessarily quasiregular, quasiregular elements need not be "bad", in a rather vague sense. In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.
Definition.
Let "R" be a ring (with unity) and let "r" be an element of "R". Then "r" is said to be quasiregular, if 1 − "r" is a unit in "R"; that is, invertible under multiplication. The notions of right or left quasiregularity correspond to the situations where 1 − "r" has a right or left inverse, respectively.
An element "x" of a non-unital ring is said to be right quasiregular if there is "y" such that formula_1. The notion of a left quasiregular element is defined in an analogous manner. The element "y" is sometimes referred to as a right quasi-inverse of "x". If the ring is unital, this definition quasiregularity coincides with that given above. If one writes formula_2, then this binary operation formula_3 is associative. In fact, the map formula_4 (where × denotes the multiplication of the ring "R") is a monoid isomorphism. Therefore, if an element possesses both a left and right quasi-inverse, they are equal.
Note that some authors use different definitions. They call an element "x" right quasiregular if there exists "y" such that formula_5, which is equivalent to saying that 1 + "x" has a right inverse when the ring is unital. If we write formula_6, then formula_7, so we can easily go from one set-up to the other by changing signs. For example, "x" is right quasiregular in one set-up iff −"x" is right quasiregular in the other set-up.
Generalization to semirings.
The notion of quasiregular element readily generalizes to semirings. If "a" is an element of a semiring "S", then an affine map from "S" to itself is formula_17. An element "a" of "S" is said to be right quasiregular if formula_18 has a fixed point, which need not be unique. Each such fixed point is called a left quasi-inverse of "a". If "b" is a left quasi-inverse of "a" and additionally "b" = "ab" + 1, then "b" it is called a quasi-inverse of "a"; any element of the semiring that has a quasi-inverse is said to be quasiregular. It is possible that some but not all elements of a semiring be quasiregular; for example, in the semiring of nonegative reals with the usual addition and multiplication of reals, formula_18 has the fixed point formula_20 for all "a" < 1, but has no fixed point for "a" ≥ 1. If every element of a semiring is quasiregular then the semiring is called a quasi-regular semiring, closed semiring, or occasionally a Lehmann semiring (the latter honoring the paper of Daniel J. Lehmann.)
Examples of quasi-regular semirings are provided by the Kleene algebras (prominently among them, the algebra of regular expressions), in which the quasi-inverse is lifted to the role of a unary operation (denoted by "a"*) defined as the least fixedpoint solution. Kleene algebras are additively idempotent but not all quasi-regular semirings are so. We can extend the example of nonegative reals to include infinity and it becomes a quasi-regular semiring with the quasi-inverse of any element "a" ≥ 1 being the infinity. This quasi-regular semiring is not additively idempotent however, so it is not a Kleene algebra. It is however a complete semiring. More generally, all complete semirings are quasiregular. The term "closed semiring" is actually used by some authors to mean complete semiring rather than just quasiregular.
Conway semirings are also quasiregular; the two Conway axioms are actually independent, i.e. there are semirings satisfying only the product-star [Conway] axiom, ("ab")* = 1+"a"("ba")*"b", but not the sum-star axiom, ("a"+"b")* = ("a"*"b")*"a"* and vice versa; it is the product-star [Conway] axiom that implies that a semiring is quasiregular. Additionally, a commutative semiring is quasiregular if and only if it satisfies the product-star Conway axiom.
Quasiregular semirings appear in s, a generalization of the shortest path problem.


Fourier transform on finite groups

In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
Definitions.
The Fourier transform of a function formula_1 at a representation formula_2 of formula_3 is
For each representation formula_5 of formula_3, formula_7 is a formula_8 matrix, where formula_9 is the degree of formula_5.
The inverse Fourier transform at an element formula_11 of formula_3 is given by
Properties.
Transform of a convolution.
The convolution of two functions formula_14 is defined as
The Fourier transform of a convolution at any representation formula_5 of formula_3 is given by
Plancherel formula.
For functions formula_14, the Plancherel formula states
where formula_21 are the irreducible representations of formula_3.
Fourier transform for finite abelian groups.
If the group "G" is a finite abelian group, the situation simplifies considerably:
The Fourier transform of a function formula_26 is the function formula_27 given by
The inverse Fourier transform is then given by
For formula_30, a choice of a primitive "n"-th root of unity formula_31 yields an isomorphism
given by formula_33. In the literature, the common choice is formula_34, which explains the formula given in the article about the discrete Fourier transform. However, such an isomorphism is not canonical, similarly to the situation that a finite-dimensional vector space is isomorphic to its dual, but giving an isomorphism requires choosing a basis.
A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply formula_35, where 0 is the group identity and formula_36 is the Kronecker delta.
Fourier Transform can also be done on cosets of a group.
Relationship with representation theory.
There is a direct relationship between the Fourier transform on finite groups and the representation theory of finite groups. The set of complex-valued functions on a finite group, formula_3, together with the operations of pointwise addition and convolution, form a ring that is naturally identified with the group ring of formula_3 over the complex numbers, formula_39. Modules of this ring are the same thing as representations. Maschke's theorem implies that formula_39 is a semisimple ring, so by the Artin–Wedderburn theorem it decomposes as a direct product of matrix rings. The Fourier transform on finite groups explicitly exhibits this decomposition, with a matrix ring of dimension formula_9 for each irreducible representation.
More specifically, the Peter-Weyl theorem (for finite groups) states that there is an isomorphism
given by
The left hand side is the group algebra of "G". The direct sum is over a complete set of inequivalent irreducible "G"-representations formula_44.
The Fourier transform for a finite group is just this isomorphism. The product formula mentioned above is equivalent to saying that this map is a ring isomorphism.
Applications.
This generalization of the discrete Fourier transform is used in numerical analysis. A circulant matrix is a matrix where every column is a cyclic shift of the previous one. Circulant matrices can be diagonalized quickly using the fast Fourier transform, and this yields a fast method for solving systems of linear equations with circulant matrices. Similarly, the Fourier transform on arbitrary groups can be used to give fast algorithms for matrices with other symmetries . These algorithms can be used for the construction of numerical methods for solving partial differential equations that preserve the symmetries of the equations .
When applied to the Boolean group formula_45, the Fourier transform on this group is the Hadamard transform, which is commonly used in quantum computing and other fields. Shor's algorithm uses both the Hadamard transform (by applying a Hadamard gate to every qubit) as well as the quantum Fourier transform. The former considers the qubits as indexed by the group formula_45 and the later considers them as indexed by formula_47 for the purpose of the Fourier transform on finite groups.


Schur orthogonality relations

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups.
They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).
Finite groups.
Intrinsic statement.
The space of complex-valued class functions of a finite group G has a natural inner product:
where formula_2 means the complex conjugate of the value of formula_3 on "g". With respect to this inner product, the irreducible characters form an orthonormal basis
for the space of class functions, and this yields the orthogonality relation for the rows of the character
table:
For formula_5, applying the same inner product to the columns of the character table yields:
where the sum is over all of the irreducible characters formula_7 of "G" and the symbol formula_8 denotes the order of the centralizer of formula_9. Note that since and are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.
The orthogonality relations can aid many computations including:
Coordinates statement.
Let formula_10 be a matrix element of an irreducible matrix representation
formula_11 of a finite group formula_12 of order |"G"|, i.e. "G" has |"G"| elements. Since it can be proven that any matrix representation of any finite group is equivalent to a unitary representation, we assume formula_11 is unitary:
where formula_15 is the (finite) dimension of the irreducible representation formula_11.
The orthogonality relations, only valid for matrix elements of "irreducible" representations, are:
Here formula_18 is the complex conjugate of formula_19 and the sum is over all elements of "G".
The Kronecker delta formula_20 is unity if the matrices are in the same irreducible representation formula_21. If formula_11 and formula_23 are non-equivalent
it is zero. The other two Kronecker delta's state that
the row and column indices must be equal (formula_24 and formula_25) in order to obtain a non-vanishing result. This theorem is also known as the Great (or Grand) Orthogonality Theorem.
Every group has an identity representation (all group elements mapped onto the real number 1).
This is an irreducible representation. The great orthogonality relations immediately imply that
for formula_27 and any irreducible representation formula_28 not equal to the identity representation.
Example of the permutation group on 3 objects.
The 3! permutations of three objects form a group of order 6, commonly denoted (symmetric group). This group is isomorphic to the point group formula_29, consisting of a threefold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irreducible representation ("l" = 2). In the case of one usually labels this representation
by the Young tableau formula_30 and in the case of formula_29
one usually writes formula_32. In both cases the representation consists of the following six real matrices, each representing a single group element:
The normalization of the (1,1) element:
In the same manner one can show the normalization of the other matrix elements: (2,2), (1,2), and (2,1).
The orthogonality of the (1,1) and (2,2) elements:
Similar relations hold for the orthogonality of the elements (1,1) and (1,2), etc.
One verifies easily in the example that all sums of corresponding matrix elements vanish because of
the orthogonality of the given irreducible representation to the identity representation.
Direct implications.
The trace of a matrix is a sum of diagonal matrix elements,
The collection of traces is the "character" formula_37 of a representation. Often one writes for
the trace of a matrix in an irreducible representation with character formula_38
In this notation we can write several character formulas:
which allows us to check whether or not a representation is irreducible. (The formula means that the lines in any character table have to be orthogonal vectors.)
And
which helps us to determine how often the irreducible representation formula_11 is contained within the reducible representation formula_43 with character formula_44.
For instance, if
and the order of the group is
then the number of times that formula_47 is contained within the given
"reducible" representation formula_43 is
"See Character theory for more about group characters."
Compact Groups.
The generalization of the orthogonality relations from finite groups to compact groups (which include compact Lie groups such as SO(3)) is basically simple: "Replace the summation over the group by an integration over the group."
Every compact group formula_50 has unique bi-invariant Haar measure, so that the volume of the group is 1. Denote this measure by formula_51. Let formula_52 be a complete set of irreducible representations of formula_50, and let formula_54 be a matrix coefficient of the representation formula_55. The orthogonality relations can then be stated in two parts:
1) If formula_56 then
2) If formula_58 is an orthonormal basis of the representation space formula_55 then
where formula_61 is the dimension of formula_55. These orthogonality relations and the fact that all of the representations have finite dimensions are consequences of the Peter–Weyl theorem.
An Example SO(3).
An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 x 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles: formula_63 (see e.g., this article for the explicit form of an element of SO(3) in terms of Euler angles). The bounds are formula_64 and formula_65.
Not only the recipe for the computation of the volume element formula_66 depends on the chosen parameters, but also the final result, i.e. the analytic form of the weight function (measure) formula_67.
For instance, the Euler angle parametrization of SO(3) gives the weight formula_68 while the n, ψ parametrization gives the weight formula_69
with formula_70
It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary:
With the shorthand notation
the orthogonality relations take the form
with the volume of the group:
As an example we note that the irreducible representations of SO(3) are Wigner D-matrices formula_75, which are of dimension formula_76. Since
they satisfy
References.
Any physically or chemically oriented book on group theory mentions the orthogonality relations. The following more advanced books give the proofs:
The following books give more mathematically inclined treatments:


Zariski ring

In commutative algebra, a Zariski ring is a commutative Noetherian topological ring "A" whose topology is defined by an ideal formula_1 contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by under the name "semi-local ring" which now means something different, and named "Zariski rings" by . Examples of Zariski rings are noetherian local rings with the topology induced by the maximal ideal, and formula_1-adic completions of Noetherian rings.
Let "A" be a Noetherian topological ring with the topology defined by an ideal formula_1. Then the following are equivalent.


Functional-theoretic algebra

Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It becomes commutative when the two functionals are the same.
Definition.
Let "AF" be a vector space over a field "F", and let "L"1 and "L"2 be two linear functionals on AF with the property "L"1("e") = "L"2("e") = 1"F" for some "e" in "AF". We define multiplication of two elements "x", "y" in "AF" by
It can be verified that the above multiplication is associative and that "e" is the identity of this multiplication.
So, AF forms an associative algebra with unit "e" and is called a "functional theoretic algebra"(FTA).
Suppose the two linear functionals "L"1 and "L"2 are the same, say "L." Then "AF" becomes a commutative algebra with multiplication defined by
Example.
"X" is a nonempty set and "F" a field. "F""X" is the set of functions from "X" to "F".
If "f, g" are in "F""X", "x" in "X" and "α" in "F", then define
and
With addition and scalar multiplication defined as this, "F""X" is a vector space over "F."
Now, fix two elements "a, b" in "X" and define a function "e" from "X" to "F" by "e"("x") = 1"F" for all "x" in "X".
Define "L"1 and "L2" from "F""X" to "F" by "L"1("f") = "f"("a") and "L"2("f") = "f"("b").
Then "L"1 and "L"2 are two linear functionals on "F""X" such that "L"1("e")= "L"2("e")= 1"F"
For "f, g" in "F""X" define
Then "F""X" becomes a non-commutative function algebra with the function "e" as the identity of multiplication.
Note that
FTA of Curves in the Complex Plane.
Let C denote the field of
Complex numbers.
A continuous function "γ" from the closed
interval [0, 1] of real numbers to the field C is called a
curve. The complex numbers "γ"(0) and "γ"(1) are, respectively,
the initial and terminal points of the curve.
If they coincide, the
curve is called a "loop".
The set "V"[0, 1] of all the curves is a
vector space over C.
We can make this vector space of curves into an
algebra by defining multiplication as above.
Choosing formula_7 we have for "α,β" in "C"[0, 1],
Then, "V"[0, 1] is a non-commutative algebra with "e" as the unity.
We illustrate
this with an example.
Example of f-Product of Curves.
Let us take (1) the line segment joining the points (1, 0) and (0, 1) and (2) the unit circle with center at the
origin.
As curves in "V"[0, 1], their equations can be obtained as
Since formula_10 the circle "g"
is a loop.
The line segment "f" starts from :formula_11
and ends at formula_12
Now, we get two "f"-products
formula_13 given by
and
See the Figure.
Observe that formula_16 showing that
multiplication is non-commutative. Also both the products starts from formula_17
References.
Book of Beautiful Curves
Certain Number Theoretic Episodes in Algebra