Abstract
ABSTRACT. We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m × n-matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These bitableaux form a Gröbner basis of J, and J has a linear minimal free resolution. These results are used to derive a new generating set for the Grothendieck group of finitely generated T m ×GL n (K)-equivariant modules over K [X]. We employ the Knuth-Robinson-Schensted correspondence and a toric deformation of the multi-Rees algebra that parameterizes the ideals J.
INTRODUCTION
Let K be a field and X an m × n matrix of indeterminates x i j over K. We write R = K [X ] for the polynomial ring in the x i j . The group GL m (K) × GL n (K) acts on R with an action induced by the rule (g, g ′ ) · X = gX g ′−1 . The representation theory of R as a module for this group is intimately connected to the linear basis of R given by bitableaux [4, Ch. 11] . The bitableaux are products of minors which are indexed by pairs of tableaux of the same shape with strictly increasing rows and weakly increasing columns. We say that a bitableau is superstandard if its left factor tableaux has column i filled with the number i. The left tableau determines the row indices of the minors whose product the bitableau represents.
INTRODUCTION
For each i, 1 ≤ i ≤ m, let J i ⊂ R denote the ideal generated by the size i minors of the first i rows of X . In the current work we study an arbitrary product of such ideals. For a decreasing sequence of positive integers min(m, n) ≥ s 1 ≥ · · · ≥ s ν we set J S = J s 1 . . . J s ν . It is a consequence of Theorem 2.2 that the ideals J S are exactly those that are generated by superstandard bitableau of shape S.
INTRODUCTION
Our main results are Theorems 3.3 and 4.7, which we summarize here as follows.
Theorem.
(1) The collection of superstandard bitableaux of shape S in R forms a Gröbner basis for the ideal J S with respect to a diagonal monomial order. ( 2) The ideal J S has a linear minimal free resolution.
Theorem.
The theorem is supplemented by results on primary decompositions and integral closedness. Statement (1) will be demonstrated in two ways. The first is via the KnuthRobinson-Schensted correspondence, and this approach, together with a brief introduction to standard bitableaux, the straightening law, and the KRS correspondence, will occupy Sections 2 and 3. The second proof of (1) and the proof of (2) are via Sagbi (or toric) deformations. It will take place in Section 4. The crucial point for (2) is that the multi-Rees algebra of the ideals J 1 , . . . , J m is a Koszul algebra, and, in its turn, this will be derived from the Koszul property of the initial algebra of the multi-Rees algebra.
Theorem.
The theorem should be viewed as occurring in the greater context of ideals generated by a family of bitableaux possessing natural Gröbner bases [2, 3, 7, 17, 19, 20] . Nevertheless, the fact that the standard bitableaux in a product ideal like J S form a Gröbner basis, is a rare phenomenon associated with ideals generated by "maximal" minors. Statement (1) of the theorem is a direct generalization of Conca's result [7] for rectangular shapes S.
Theorem.
In Section 5 we use statement (1) of the theorem to derive a new generating set for the Grothendieck group of finitely generated T m × GL n (K)-equivariant R-modules, where T m ⊂ GL m (K) is the torus of diagonal matrices. Having a basis for this group coming from structure sheaves of schemes was the original motivation for studying the class of ideals J S .
THE STRAIGHTENING LAW
Let K be a field and X = (x i j ) an m × n matrix of indeterminates x i j over K. We will study determinantal ideals in the polynomial ring
THE STRAIGHTENING LAW
Almost all of the approaches one can choose for the investigation of determinantal ideals use standard bitableaux and the straightening law. The principle governing this approach is to consider all the minors of X (and not just the 1-minors x i j ) as generators of the K-algebra R so that products of minors appear as "monomials". The price to be paid, of course, is that one has to choose a proper subset of all these "monomials" as a linearly independent K-basis: the standard bitableaux to be defined below are a natural choice for such a basis, and the straightening law tells us how to express an arbitrary product of minors as a K-linear combination of the basis elements. (In [4] standard bitableaux were called standard monomials; however, we will have to consider the ordinary monomials in K[X ] so often that we reserve the term "monomial" for products of the x i j .)
THE STRAIGHTENING LAW
In the following
THE STRAIGHTENING LAW
stands for the determinant of the submatrix (x a i b j : i = 1, . . . ,t, j = 1, . . .,t). The letter ∆ always denotes a product δ 1 · · · δ w of minors, and we assume that the sizes |δ i | (i. e. the number of rows of the submatrix X ′ of X such that δ i = det(X ′ )) are descending, |δ 1 | ≥ · · · ≥ |δ w |. By convention, the empty minor [ | ] denotes 1. The shape |∆| of ∆ is the sequence (|δ 1 |, . . ., |δ w |). If necessary we may add factors [ | ] at the right hand side of the products, and extend the shape accordingly.
THE STRAIGHTENING LAW
A product of minors is also called a bitableau. The choice of this term "bitableau" is motivated by the graphical description of a product ∆ as a pair of Young tableaux as in Figure 1 : Every product of minors is represented by a bitableau and, conversely, every bitableau stands for a product of minors if the length of the rows is decreasing from top to bottom, the entries in each row are strictly increasing from the middle to the outmost box, the entries of the left tableau are in {1, . . ., m} and those of the right tableau are in {1, . . ., n}. These conditions are always assumed to hold.
THE STRAIGHTENING LAW
For formal correctness one should consider the bitableaux as purely combinatorial objects and distinguish them from the ring-theoretic objects represented by them, but since there is no real danger of confusion, we simply identify them.
THE STRAIGHTENING LAW
Whether ∆ is a standard bitableau is controlled by a partial order of the minors, namely Let e 1 , . . . , e m and f 1 , . . . , f n denote the canonical Z-bases of Z m and Z n respectively. Clearly K[X ] is a Z m ⊕ Z n -graded algebra if we give x i j the "vector bidegree" e i ⊕ f j . All minors are homogeneous with respect to this grading. In a bitableau of bidegree (c 1 , . . ., c m , d 1 , . . . , d n ) ∈ Z m ⊕Z n , row i appears with multiplicity c i , and column j appears with multiplicity d j , i = 1, . . . , m, j = 1, . . ., n. The straightening relations must therefore preserve these multiplicities, whose collection is often called the content of the bitableau.
THE STRAIGHTENING LAW
We say that an ideal I ⊂ R has a standard basis if I is the K-vector space spanned by the standard bitableaux Σ ∈ I. Let S = s 1 , . . ., s v be weakly decreasing sequence of positive integers s i ≤ min(m, n). In this article we investigate the ideal
THE STRAIGHTENING LAW
where J t is the ideal generated by the t-minors of the first t rows of X . In other words, J t is the the ideal of maximal minors of the matrix X t formed by the first t rows of X in K[X t ] and extended to K[X ]. We will see that the ideals J S behave very much like the powers of ideals of maximal minors that they generalize in a natural way.
THE STRAIGHTENING LAW
The bitableaux ∆ = δ 1 · · · δ v with δ i ∈ J s i , |δ i | = s i , are automatically standard on the left side (the tableau of row indices). We call them row superstandard and just superstandard if they are also standard on the right side. Note that in a (row) superstandard bitableau all indices a i j are as small as possible, namely a i j = j. In [4] superstandard tableaux are called row initial, but we want to reserve the term "initial" for use in connection with monomial orders.
THE STRAIGHTENING LAW
Let
THE STRAIGHTENING LAW
. . , w. Subtableaux of (super)standard bitableaux are evidently (super)standard.
Theorem 2.2. The ideal J S has a standard basis that is given by all standard bitableaux containing a superstandard tableau of shape S.
Proof. As a vector space over K, J S is certainly generated by all products
Theorem 2.2. The ideal J S has a standard basis that is given by all standard bitableaux containing a superstandard tableau of shape S.
(We do not assume that the δ i are ordered by size.) It is enough to show that this property is preserved by all products of minors that arise if we replace an incomparable subproduct δ i δ j by the right hand side of the straightening relation. After finitely many steps we arrive at a K-linear combination of standard bitableaux, each of which contains a superstandard tableau of shape S.
Theorem 2.2. The ideal J S has a standard basis that is given by all standard bitableaux containing a superstandard tableau of shape S.
The description of the standard basis yields the primary composition of the ideals J S as an easy consequence:
Theorem 2.2. The ideal J S has a standard basis that is given by all standard bitableaux containing a superstandard tableau of shape S.
Proof. The ideals on both sides have a standard basis as follows from the theorem. Therefore it is enough to compare these. But a standard bitableau contains a superstandard bitableau of shape S if and only if it contains a rectangular superstandard bitableau with e i rows of length t i for every i, and the latter form the standard basis of J t i arises from I t i (X t i ) e i by tensoring over K with the polynomial ring in the variables x kl outside X t i , and such extensions preserve the property of being primary. That the powers of I t i (X t i ) are primary is well-known; see [4, 9.18] .
Theorem 2.2. The ideal J S has a standard basis that is given by all standard bitableaux containing a superstandard tableau of shape S.
For the last statement it is enough to note that the powers J e i t i are not only primary, but also integrally closed. This follows from the normality of the Rees algebra R(
Theorem 2.2. The ideal J S has a standard basis that is given by all standard bitableaux containing a superstandard tableau of shape S.
The statement on integral closedness is equivalent to the normality of a multi-Rees algebra. We postpone this aspect until Theorem 4.7.
THE KNUTH-ROBINSON-SCHENSTED CORRESPONDENCE
Let Σ be a standard bitableau. The Knuth-Robinson-Schensted correspondence KRS (see Fulton [9] or Stanley [16] ) sets up a bijective correspondence between standard bitableaux and monomials in the ring K[X ]. The treatment of KRS below follows [2] and [3] . However, for better compatibility with the definition of the ideals J S we have exchanged the roles of the left and right tableau.
THE KNUTH-ROBINSON-SCHENSTED CORRESPONDENCE
If one starts from bitableaux, the correspondence is constructed from the algorithm KRS-step [3, 4.2] (based on deletion [3, 4.1] ). Let Σ = (a i j |b i j ) be a non-empty standard bitableau. The output of KRS-step is a triple (Σ ′ , ℓ, r) consisting of a standard bitableau Σ ′ and a pair of integers (ℓ, r) constructed as follows.
THE KNUTH-ROBINSON-SCHENSTED CORRESPONDENCE
(a) One chooses the largest entry r in the right tableau of Σ; suppose that procedure on the left tableau as follows: (i) if p = 1, then ℓ = s is the second component of the output, and the first is the standard bitableau Σ ′ that has now been created; (ii) otherwise s is moved one row up and pushes out the left most entry a p−1k such that a p−1k ≤ s whereas a p−1k is stored in s. (iii) one replaces p by p − 1 and goes to step (i).
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
There is an inverse to deletion, called insertion that can be easily constructed by inverting all steps in deletion. Together they prove the main theorem on KRS: For insertion one must order the factors of a monomial in a way that respects the monotonicity properties of KRS-step: let x r 1 ℓ 1 · · · x r k ℓ k = KRS(Σ) with the factors ordered as in the definition of KRS; then r i ≤ r i+1 and
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
See [3, p. 37] (with r and ℓ exchanged). Property ( * ) allows us to take care of a superstandard subtableau, but some additional bookkeeping is necessary. To this end we extend the output of KRS-step by a further component ρ, the row mark that we will now define.
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
(Here "row" refers to the tableau, not to a minor.) Let S = s 1 , . . . , s v a nonincreasing sequence as above, and suppose that Σ contains a superstandard bitableau of shape S. Then we can distinguish boxes in the left tableau that belong to the superstandard bitableau from those that do not belong to it, namely the box at position (i, j) belongs to the superstandard subtableau if and only if a i j = j and j ≤ s i . We supplement step (d) above by (iv) if a i j = j and j ≤ s i , but (i, j) is the pivot position or a ′ i j > a i j , then ρ = i is the fourth component of the output of KRS-step. Otherwise we set ρ = 0.
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
Let us first make sure that rule (iv) makes sense by showing that there can be at most one row i with a i j = j and a ′ i j > a i j . This is clear if (i, j) is the pivot position since all remaining positions remain unchanged. In the other case, if a i j = j and a ′ i j > a i j , then i = max{k : a k j } = j. In fact, if the box at position (i, j) is hit by the push out sequence in KRS-step(d) and a i j = j, then the entry j is pushed out into the next upper row and replaces a i−1 j = j by j.
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
The triples (ℓ, r, ρ) form the columns of a three row array krs(Σ) that we build by listing the triples (ℓ, r, ρ) from right to left as follows:
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
We give an example in Figure 2 with S = 3, 2. The circles in the right tableau mark the pivot position, those in the left mark the chains of "pushouts":
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
The three row array produced by the example of is the KRS image of a superstandard bitableau of shape (3, 2) (though it is not the KRS image of the superstandard subtableau contained in Σ). What we have observed in this special case is always true, as we will be stated in Lemma 3.2 below. Let
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
be the product of the indeterminates in the main diagonal of [a 1 . . .
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
It is easy to see that the map diag is not injective on standard bitableaux (let alone all bitableaux), in contrast to KRS. (Otherwise KRS would be completely superfluous in the study of determinantal ideals.) However, if Σ is a superstandard bitableau, then
It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(Σ)
since the whole push out sequence in KRS-step(d) always replaces the entry of a box by itself.
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
Proof. Suppose T is a (not necessarily standard) bitableau whose row tableau is superstandard of shape S. Then diag(T) = diag(T ′ ) where T ′ is standard of shape S. This is easy to see and left to the reader. Therefore it is enough to prove the lemma without the requirement that T is standard. (Equation (3.1) would allow us to replace diag(T) by KRS(T), but this is irrelevant.) Let Σ = (a i j | b i j ) and choose an index k such that row k of Σ occurs in the superstandard subtableau. Let s = max{ j : a k j = j}. As in the example we extract the subarray A from krs(Σ) with row mark k. We claim that the corresponding monomial is the diagonal of an s-minor [1 . . . s | c 1 . . . c s ]. This claim amounts to the following conditions for the subarray A:
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
(1) The entries of the first row are 1, . . ., s in ascending order; (2) the entries c 1 , . . . , c s of the second row are strictly increasing. First of all we note that the row mark is k if a box at position (k, z) with a kz = z changes its content in KRS-step: either (k, z) is the pivot position or in
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
This change happens exactly once for j = 1, . . ., s. Therefore the entries of the first row of A are indeed 1, . . . , s.
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
But 1, . . . , s are also produced in the right order. If a kz = z, then a kw = w for w = 1, . . . , z − 1, and so these boxes have not yet changed content. Moreover the component r of the output of KS-step is exactly z, and 1, . . . , z − 1 will be produced later. This proves (1).
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
The entries c 1 , . . . , c s in the second row are automatically weakly increasing by the first inequality in ( * ), and an equality of two entries would contradict (1) because of the second inequality in ( * ). In other words, (1) implies (2).
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
It is now time to introduce a diagonal monomial (or term) order ≺ on the polynomial ring K[X ]. This is a term order on the polynomial ring under which the initial monomial of each minor is the product of the elements in the main diagonal:
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
Diagonal monomial orders are the standard choice in the study of determinantal ideals from the Gröbner basis viewpoint. See [3] for a survey that also contains a brief introduction to general Gröbner bases and initial ideals. (1) the row superstandard bitableaux of shape S form a Gröbner basis of J S . Since ∏ in ≺ (J s i ) ⊂ in ≺ (J S ) for obvious reasons, it is enough to observe the converse for (3) . But this follows again from Lemma 3.2 since in ≺ (T) is contained in ∏ in ≺ (J s i ).
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
In the terminology of [2] In this section we will use Sagbi deformations of Rees algebras to study the ideals J S defined in the previous sections. By definition, these ideals are products of powers of the ideas J 1 , . . ., J m (we do not assume that n ≥ m; if m > n then all results in this section hold with J n+1 = 0, . . ., J m = 0.)
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
Before we turn to our class of ideals we study the Sagbi approach via Rees algebras in general. Let A = K[x 1 , . . . , x r ] the polynomial ring in r indeterminates, endowed with a monomial order ≺. For every K-vector subspace V of A we may consider the vector space in ≺ (V ) generated by the monomials in ≺ ( f ) as f = 0 varies in V . If V is an ideal of A, then in ≺ (V ) will be an ideal of A, and if V is a K-subalgebra of A, then in ≺ (V ) will be a K-subalgebra of A as well. If V is an ideal, then a subset G of V is a Gröbner basis if in ≺ (V ) is generated (as an ideal) by {in ≺ ( f ) : f ∈ G}. Similarly, if V is an algebra, then a subset G of V is a Sagbi basis if in ≺ (V ) is generated (as a K-algebra) by {in ≺ ( f ) : f ∈ G}. A variation of the Buchberger criterion allows us to detect whether a given set G of polynomials is a Sagbi basis. One has to replace the so called S-pairs with the binomial relations defining the toric ring K[in ≺ ( f ) : f ∈ G]. We refer the reader to [6] for further details.
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
Let now I 1 , . . ., I v homogeneous ideals of A. We want to express the condition
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
in terms of Sagbi deformations. Let
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
be the (multi-)Rees ring R(I 1 , . . . , I v ) associated to the family I 1 , . . ., I v . In order to describe it as a as a subalgebra of a polynomial ring, we take new variables y 1 , . . . , y v . Then we can identify R(I 1 , . . . , I v ) with the subalgebra
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
By construction, R(I 1 , . . . , I v ) has a Z ⊕ Z v -graded structure induced by the assignment deg(x i ) = e 0 for all i and deg(y j ) = e j for all j where e 0 , e 1 , . . . , e v denotes the canonical basis of Z ⊕ Z v . We extend ≺ to a monomial order on K [x, y] . It is indeed irrelevant which extension is chosen because the polynomials we will consider are "monomial" in the y's and so we denote the extension by ≺ as well. To test whether condition (4.3) holds we can use the Sagbi variant of the Buchberger criterion [6] . Set
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
and consider two A-algebra maps from the polynomial ring
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
to A[y] defined as follows:
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
By construction
Lemma 3.2. Let Σ be a standard bitableau containing a superstandard bitableau of shape S. Then there exists a superstandard bitableau T of shape S such that diag(T) divides KRS(Σ).
The kernel of Φ is a toric ideal, i.e., a prime ideal generated by binomials since R(in ≺ (I 1 ), . . . , in ≺ (I v )) is a K-algebra generated by monomials. These binomials replace the S-pairs in the Buchberger criterion for Gröbner bases. Roughly speaking, the following criterion says that every such binomial relation of the initial monomials can be "lifted" to a relation of the elements of G themselves.
Lemma 4.1 (Sagbi version of the Buchberger criterion)
. Let G be a set of binomials generating Ker Φ. Suppose that for every g ∈ G such that Ψ(g) = 0 one has:
Lemma 4.1 (Sagbi version of the Buchberger criterion)
Then X ∪ F is a Sagbi basis.
Remark 4.2.
If g has total degree 1 in the p i j 's, then the condition required in Lemma 4.1 is automatically satisfied because F i1 , . . . , F ic i is a Gröbner basis of the ideal I i . So we have only to worry about the g ∈ G of degree > 1 in the p i j 's.
Remark 4.2.
Assume now that each ideal I i is generated in a single degree, say d i . Then R(I 1 , . . . , I v ) can be given the structure of a standard Z ⊕ Z v -graded K-algebra by assigning the degree e j − d j e 0 to y j , j = 1, . . ., v, and e 0 to the variables x i . On P we define the grading by deg(x i ) = e 0 and deg(p i j ) = e i . Then the maps Φ and Ψ are Z ⊕ Z v -graded.
Remark 4.2.
The following theorem relates a ring theoretic property of the Rees algebra to the free resolutions of the ideals involved: 
Remark 4.2.
The set A inherits the partial order from the set of all minors that has been introduced for the straightening law (see Section 2) . The set of all minors is a distributive lattice with respect to this order, and A is a sublattice: suppose that r ≤ s; to wit, Proof. It is enough to prove that the given elements are a Gröbner basis of Ker Φ . The argument is quite standard (for example, see for instance [18, Chap.14] for similar statements) and so we just sketch it. First note that a monomial order selecting the underlined monomials is given by taking the reverse lexicographic order associated to a total order on the p a 's that refines the partial order A .
Remark 4.2.
To prove the assertion we choose an arbitrary monomial in the image of Φ, say [3, 3.12] and apply [18, Prop.13.15] which implies that in ≺ (R(J 1 , . . . , J m )) is normal since its defining ideal has a square-free initial ideal.
EQUIVARIANT R-MODULES
In this section we make the assumption that m ≥ n. This will simplify the conclusion of main result of the section, which has a less pleasing analogue when m < n.
EQUIVARIANT R-MODULES
Let T m ⊂ GL m (K) denote the diagonal torus, and set G := T m × GL n (K). Then G acts on R as in Section 1. In this section we consider the Grothendieck group of finitely generated G-equivariant R-modules with a rational G-action, denoted K 0 G (R). Since R is a polynomial ring, the group K 0 G (R) can be identified with the representation ring of G. Hence K 0 G (R) is generated by the free equivariant modules R ⊗V , as V ranges over all finite dimensional rational G modules. The group K 0 G (R) inherits a product from the tensor product of G-modules. The product of the classes of two general equivariant R-modules can be expressed in terms of their Tor-modules, a fact we will not need here.
EQUIVARIANT R-MODULES
Using the multigrading of Section 2, an equivariant R-module M is at once seen to be a multigraded module. We write its Hilbert series as
EQUIVARIANT R-MODULES
Here the group S n is permuting the v variables, and the GL n (K) The superstandard bitableau of shape S span a representation of G [4, Thm. 11.5(a)]. It follows that the ideals J S are G-invariant, and hence the quotient ring R/J S defines an element of K 0 G (R). This stands in contrast to an ideal generated by standard bitableaux with a fixed left tableau, which does not necessarily a G-invariant ideal (see [4, Rmk. 11.12] ). For any shape S whose part sizes are at most n, let σ S (v) denote the Schur polynomial in variables v 1 , . . ., v n . That is, σ S (v) will be the generating function in v for the content of tableaux of shape S with strictly increasing rows, weakly increasing columns and entries in {1, . . ., n}.
EQUIVARIANT R-MODULES
The ideal J S is generated by an irreducible representation of G whose character is u To finish the proof, we must show that every Schur polynomial can be written as a finite

{'paper_id': '45860681', 'title': 'Ideals generated by superstandard tableaux', 's2_url': 'https://api.semanticscholar.org/CorpusID:45860681', 'source': 'S2ORC'}