Bases for Spaces of Highest Weight Vectors in Arbitrary Characteristic

Let k be an algebraically closed field of arbitrary characteristic. First we give explicit bases for the highest weight vectors for the action of GLr ×GLs on the coordinate ring k[Matrsm]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k[\text {Mat}_{rs}^{m}]$\end{document} of m-tuples of r × s-matrices. It turns out that this is done most conveniently by giving an explicit good GLr ×GLs-filtration on k[Matrsm]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k[\text {Mat}_{rs}^{m}]$\end{document}. Then we deduce from this result explicit spanning sets of the k[Matn]GLn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k[\text {Mat}_{n}]^{\text {GL}_{n}}$\end{document}-modules of highest weight vectors in the coordinate ring k[Matn] under the conjugation action of GLn.


Introduction
Let k be an algebraically closed field, let GL n be the group of invertible n × n matrices with entries in k and let T n and U n be the subgroups of diagonal matrices and of upper unitriangular matrices respectively. The group GL r × GL s acts on the k-vector space Mat Our first goal in this paper is to give bases of the vector spaces k[Mat m rs ] U r ×U s (μ,λ) . In [15] this was done under the assumption that k is of characteristic 0. The method there was to reduce the problem via a few simple isomorphisms to certain results from the representation theory of the symmetric group which were originally due to J. Donin. Although this method is rather straightforward, it is hard to generalise to arbitrary characteristic. In the present paper we solve the problem in arbitrary characteristic using results on bideterminants from the work of Kouwenhoven [11] which is based on work of Clausen [2,3]. We introduce "twisted bideterminants" to construct an explicit "good" filtration and, in particular, give bases for the spaces of highest weight vectors in k[Mat m rs ], see Theorem 1 and its two corollaries in Section 3. It turns out that these bases can also be obtained by dividing the basis elements from [15,Thm. 4] by certain integers in the obvious Z-form and then reducing mod p.
As an application we give in Section 4 explicit finite homogeneous spanning sets of the k[Mat n ] GL n -modules of highest weight vectors in the coordinate ring k[Mat n ] under the conjugation action of GL n , see Theorem 4 in Section 4. Although this problem is difficult to tackle directly, we gave in [15] a method in arbitrary characteristic called "transmutation" to reduce this problem to giving spanning sets for the vector spaces k[Mat m rs ] U r ×U s (μ,λ) , see Theorem 2 in the present paper. So the problem is reduced to the problem we solved in Section 3.

Preliminaries
The field k, the groups GL n , U n , T n , the variety Mat m rs and its coordinate ring k[Mat m rs ] are as in the introduction. Note that k[Mat m rs ] is the polynomial algebra over k in the variables x(l) ij , 1 ≤ l ≤ m, 1 ≤ i ≤ r, 1 ≤ j ≤ s, where x(l) ij is the function which assigns the entry in the i-th row and j -th column of the l-th matrix. If m = 1 we write x ij instead of x(1) ij . The GL r × GL s -module k[Mat m rs ] is multigraded by tuples of integers ≥ 0 (not necessarily partitions) of length m. We denote the set of such tuples with coordinate sum t by m,t or just t . In this section we will only consider the GL r × GL s -module k[Mat rs ], although we will use the set m,t .

Skew Young Diagrams and Tableaux
In this section we introduce some combinatorics that we will need in Section 3 and which originates from [13,16,17], and [6,7]. In Section 2.2 we discuss interpretations in terms of representation theory.
For λ a partition of n we denote the length of λ by l(λ) and its coordinate sum by |λ|. We will identify each partition λ with the corresponding Young diagram The (i, j ) ∈ λ are called the boxes or cells of λ. More generally, if λ, μ are partitions with λ ⊇ μ, then we denote the diagram λ with the boxes of μ removed by λ/μ and call it the skew Young diagram associated to the pair (λ, μ). Of course the skew diagram λ/μ does not determine λ and μ. For a skew diagram E, we will denote the transpose by E and the number of boxes by |E|. The group of permutations of the boxes of E will be denoted by Sym(E), and the column stabliser of E in Sym(E), that is, the product of the groups of permutations of each column of E, will be denoted by C E . By diagram mapping we mean a bijection between two diagrams as subsets of N × N.
Let E be a skew diagram with t boxes. A skew tableau of shape E is a mapping T : E → N = {1, 2, . . .}. A skew tableau of shape E is called ordered if its entries are weakly increasing along rows and weakly increasing down columns, and it is called semi-standard if its entries are weakly increasing along rows and strictly increasing down columns. It is called a t-tableau if its entries are the numbers 1, . . . , t (so the entries must be distinct). A t-tableau whose entries are strictly increasing both along rows and down columns is called standard. If m is the biggest integer occurring in a tableau T , then the weight of T is the m-tuple whose i-th component is the number of occurrences of i in T . Sometimes we will also consider the weight of T as an m -tuple for some m ≥ m by extending it with zeros.
For a skew shape E with t boxes, we define the canonical skew tableau S E by filling the boxes in the i-th row with i's, and we define the tableau T E by filling in the numbers 1, . . . , t row by row from left to right and top to bottom. So S E is semi-standard, and T E is a t-tableau which is standard. The standard enumeration of a tableau T of shape E is the t-tuple obtained from T by reading its entries row by row from left to right and top to bottom.
Let μ be the tuple of row lengths of E, i.e. the weight of S E . Let S be a tableau of shape F and weight μ. If S = S E • α for some diagram mapping α : F → E, then we say that α represents S. We call S (E-)special if it is semi-standard and has a representative α : F → E such that for any a, b ∈ F , if α(b) occurs strictly below α(a) in the same column, then b occurs in a strictly lower row than a. We call a diagram mapping α : F → E admissible if for α(b) strictly below α(a) in the same column, b occurs in a strictly lower row than a and in a column to the left of a or in the same column.
Define two orderings ≤ and on N × N as follows: (p, q) ≤ (r, s) if and only if p ≤ r and q ≤ s, and (p, q) (r, s) if and only if p < r or (p = r and q ≥ s). Note that is a linear ordering. Recall that skew Young diagrams are by definition subsets of N × N. A diagram mapping α : F → E is called special if α : (F, ≤) → (E, ) and α −1 : (E, ≤) → (F, ) are order preserving. So α is special if and only if α −1 is special.
Example 2.1 Let F = (2, 2) and E = (3, 2)/(1) be skew diagrams. Since each has four boxes, we can construct a diagram mapping between the two shapes. Give F the standard enumeration. We now define a diagram mapping α 1 : F → E by numbering the boxes of E: a ∈ F is mapped by α 1 to the box of E which has the same number. Then α 1 is not admissible, since 4 is below 1 in the same column of E, but it occurs in a column strictly to the right of 1 in F . We now form the canonical tableau S E on E and pull this numbering back to F via α 1 to obtain the tableau S = S E • α 1 : Clearly S is semi-standard and for all a, b ∈ F , b occurs in a strictly lower row than a whenever α 1 (b) occurs strictly below α 1 (a) in the same column. So S is E-special semistandard. Now define α 2 , α 3 : F → E by: Then S E •α 2 = S E •α 3 = S, α 2 is admissible, but not special, and α 3 is special. The inverse of α 3 is also special and is therefore the unique special representative of the F -special semi-standard tableau T = S F • α −1 3 on E: Besides T there is one other semi-standard tableauT of shape E and weight (2, 2): This tableau is not F -special: if β : E → F is a diagram mapping withT = S F • β and b is the rightmost box in the top row of E, then there must be a box a of E such that β(a) is directly above β(b) in the same column in F , but a cannot occur in a higher row than b in E.
From now on we will always insist that representatives of special semi-standard tableaux are admissible.
What we will be using in the proof of our main result Theorem 1 is a refinement of the above combinatorics. We need to cut F and E into pieces labelled by certain integers and then we work with certain diagram mappings α which map each piece of F into the piece of E of the same label. We then apply the above combinatorics to the restrictions of α to these pieces. Now let E and F be arbitrary skew diagrams each with t boxes. Let P and Q be ordered tableaux of shapes E and F , both of weight ν ∈ t . Then a diagram mapping α : is the restriction of α to Q −1 (i). We will say that α represents (*). Notice that the m-tuples (*), for varying α, all have the same tuple of shapes and the same tuple of weights. We express this by saying that the tuple of tableaux has shapes determined by Q and weights determined by P . When the tableaux S P −1 (i) • α i are special semi-standard, we require the α i to be admissible. For more detail see [15,Sect. 3], or [16,17] where special diagram mappings are defined as "pictures". Then α 1 goes between the "1-pieces" of Q and P and α 2 goes between the "2-pieces" of Q and P . We also indicate the canonical numberings on the pieces of E and certain special semi-standard numberings on the pieces of F which can be obtained by pulling back the canonical numberings along suitable α i . is the only other P −1 (2)-special semi-standard tableau of shape Q −1 (2) and weight (2, 1, 3).

Bideterminants and Skew Schur and Specht Modules
In this section we will review some facts from the representation theory of the general linear group as well as the symmetric group. Bideterminants are introduced in [5] and their skew versions in [1]. Other sources are [2-4, 10, 11]. In the latter three their application to the representations of the symmetric group is also discussed. The representation theory of the symmetric group will not be used in this paper, but it may help to understand the combinatorics we use. It was also used in [15,Thm. 4] to obtain a version in characteristic 0 of Corollary 2 to Theorem 1 from the present paper. Let E be a skew diagram with t boxes. Let S and T be tableaux of shape E, S with entries ≤ r and T with entries ≤ s. Then we define the bideterminant where E i is the i-th column of E and n is the number of columns in E. Note that we have where C E ≤ Sym(E) is the column stabiliser of E.
As is well-known, the elements (S | T ), S standard with entries ≤ r and T standard with entries ≤ s form a basis of k[Mat rs ], see [5]. In fact one can use bideterminants to construct explicit "good" filtrations of k[Mat rs ] as a GL r × GL s -module, see [4].
Drop for the moment the assumption that k is algebraically closed. The skew Specht module S(E) = S t (E) = S t,k (E) for the group algebra A = A t,k = kSym t of the symmetric group Sym t on {1, . . . , t} is defined just as in the case of an ordinary Young diagram: S(E) = Ae 1 e 2 , where e 1 is the column anti-symmetriser of T E and e 2 is the row symmetriser of T E . The module M(E) = M t,k (E) = Ae 2 is called the permutation module associated to E. Let E, F , μ be as in Section 2.1. If k has characteristic 0, then the number of special semi-standard tableaux of shape F and weight μ is equal to the dimension of where N Sym t denotes the space of coinvariants of an A-module N , i.e. the quotient of N by the span of the elements The skew Schur module associated to a shape E, denoted by ∇ GL r (E), is the span in k[Mat rs ], s ≥ the number of rows of E, of all the bideterminants (S | S E ) where S is a tableau of shape E and with entries ≤ r. The skew Schur module ∇ GL r (E) will be nonzero if and only if r is ≥ the length of each column of E. It can easily be seen that ∇ GL r (E) is GL r -stable, and it is well-known that the set of bideterminants (S | S E ) with S as above and in addition semi-standard form a basis. Note that if E is an ordinary Young tableau then ∇ GL r (E) is the Schur (or induced) module associated to it. Now one can redefine S t,k (E) as the weight space ∇ GL r (E) 1 t for any r ≥ t. Note that this weight space is indeed stable under Sym t ≤ GL r , where Sym t is identified with the group of permutation matrices whose nonzero off-diagonal entries are restricted to the first t rows (and columns).
The co-Schur or Weyl module GL r (E) associated to a shape E can be defined as the , where g is the transpose of g ∈ GL r . For more details we refer to [1,10,11].
If E, F , μ are as in Section 2.1, then the number of special semi-standard tableaux of shape F and weight μ is equal to dim Hom GL r ( GL r (F ), ∇ GL r (E)) whenever r is ≥ the number of rows of E or ≥ the number of rows of F . This can be seen by reducing to the case that k has characteristic 0 using that GL r (F ) has a Weyl filtration, ∇ GL r (E) has a good filtration and [14,Prop. II.4.13], and then using standard properties of skew Schur functions. 1 Assume r = r 1 + · · · + r m for certain integers r i > 0, let ν ∈ m,t and let Sym ν ≤ Sym t be the Young subgroup associated to ν. If k has characteristic 0, then we have an isomorphism S t (E) ∼ = P m i=1 S ν i (P −1 (i)) of Sym ν -modules, where the sum is over all ordered tableau P of shape E and weight ν. For k arbitrary, there exists a m i=1 GL r imodule filtration of the piece of multidegree ν of ∇ GL r (E) with sections in some order isomorphic to the modules m i=1 ∇ GL r i P −1 (i) , P an ordered tableau of shape E and weight ν. Here we can omit the P 's for which P −1 (i) has a column of length > r i for some i. See [1,Thm. II.4.11] or [11,Thm. 1.4] and Remark 2 after it.
where T is a tableau of shape E and with entries ≤ r.
Then the action of GL r comes from the right multiplication rather than from the left multiplication. 2. Let λ and μ be partitions with μ ⊆ λ. Let r, r 1 , s be integers ≥ 0 with r 1 , s ≥ l(λ) and r 1 ≥ l(μ) + r and put r = r 1 − r. We embed GL r × GL r in GL r 1 such that GL r fixes the first r standard basis vectors. Then one can embed ∇ GL r (λ/μ) as a GL r -submodule in ∇ GL r 1 (λ). Indeed one can deduce from [8] that where μ is considered as a weight for T r . One can also construct an explicit isomorphism as follows. Let E ∈ Mat r s be the matrix whose first min(r , s) rows are those of the s × s identity matrix followed by r − s zero rows if r > s. Then the comorphism of the morphism A → E A : Mat rs → Mat r 1 s maps ∇ GL r 1 (λ) U r isomorphically onto ∇ GL r (λ/μ). Combinatorially this is easy to understand: ∇ GL r 1 (λ) U r has a basis labelled by semi-standard tableaux of shape λ with entries ≤ r 1 in which the entries ≤ r occupy the boxes of μ and form the canonical tableau S μ . These tableaux are clearly in one-one correspondence with the semi-standard tableaux of shape λ/μ with entries ≤ r: just remove the μ-part and subtract r from the entries of the resulting tableau of shape λ/μ. . In [15,Sect. 3] we worked with S(μ) and S(λ) which can be thought of as spanned by bideterminants (T | S μ ), T a t-tableau of shape μ, and (T | S λ ), T a t-tableau of shape λ. Actually, we mostly worked with skew versions of S(μ) and S(λ). Only after [15,Prop. 3] we passed to coinvariants. In the present paper we work entirely inside the space of coinvariants which is the degree ν piece of k[Mat m rs ] (μ,λ) . This means that t-tableaux play almost no role, they are "replaced" by diagram mappings α : μ → λ. The canonical tableaux S μ and S λ are now arbitrary tableaux S and T of shape μ and λ and we work with twisted bideterminants (S | α T ).

The Action of GL r × GL s on Several r × s-matrices
Let λ, μ be partitions of t with l(μ) ≤ r and l(λ) ≤ s, let P , Q ordered tableaux of shapes λ and μ, both of weight ν ∈ t and α : μ → λ a diagram mapping such that P • α = Q.
where b 1 is the row index of a box b. It was proved in [15,Thm. 4] that for suitable (ν, P , Q, α) these elements form a basis of the vector space k[Mat m rs ] U r ×U s (μ,λ) when k has characteristic 0. 2 More generally, we can consider for E and F skew shapes with t boxes, P , Q tableaux of shapes E and F , both of weight ν ∈ t , α : F → E a diagram mapping such that P • α = Q, S a tableau of shape F with entries ≤ r and T a tableau of shape E with entries ≤ s the sum (π,σ )∈C F ×C E sgn(π )sgn(σ ) a∈F x(Q(a)) S(π(a)), T (σ (α(a))) . (2) Note that we obtain (1) from (2) by taking S and T the canonical tableaux S F and S E .
We will now show that, when k has characteristic 0, (2) is in Z[Mat m rs ] = Z[(x(l) ij ) lij ] divisible by the order of the subgroup In each of the two lines above one may omit "(Q)" in C F (Q) or "(P )" in C E (P ), but not both. (2) only depends on the left coset of (π, σ ) modulo C P ,Q,α .

We now define the twisted bideterminant (S | m α T ) ∈ k[Mat m rs ] by
x(Q(a)) S(π(a)), T (σ (α(a))) , where the sum is over a set of representatives of the left cosets of C P ,Q,α in C F × C E .
In case m = 1, P and Q are constant equal to 1 and they play no role. We then omit P , Q and the superscript m in our notation and instead of x(1) ij we write x ij . So (S | α T ) = (π,σ ) sgn(π )sgn(σ ) a∈F x S(π(a)), T (σ (α(a))) , (4) where the sum is over a set of representatives of the left cosets of Note that if m = 1, E = F and α = id we get the ordinary bideterminant.
Remark 2 If X is a set of representatives for the left cosets of αC F (Q)α −1 ∩ C E (P ) in C E , then C F × X is a set of representatives for the left cosets of C P ,Q,α in C F × C E . If we concatenate all matrices in an m-tuple column-wise, then we obtain an isomorphism k[Mat m rs ] ∼ = k[Mat r,ms ] which maps x(l) ij to x i,(l−1)s+j . Now we have where T α,σ (a) = T (σ (α(a))) + (Q(a) − 1)s for a ∈ F . Of course we could also work with a setX of representatives for the left cosets of Then the above sum would be over σ ∈X with T α,σ (a) = T (α(σ (a))) + (Q(a) − 1)s for a ∈ F . Similarly, if X is a set of representatives for the left cosets of C F (Q) ∩ α −1 C E (P )α in C F , then X × C E is a set of representatives for the left cosets of C P ,Q,α in C F × C E .
In the case of the twisted bideterminants (S | α T ) for a single matrix, P and Q play no role, so C F (Q) and C E (P ) can be replaced by C F and C E , and in the definitions of T α,σ and S α,π the terms containing Q or P should be omitted. The twisted bideterminants (S | α T ) are known as "shuffle-products", and moving from the single matrix version of the first expression above to that of the second is called "overturn of the P-shuffle product onto the L-side", see [3,. For S a tableau of shape F with entries ≤ 2, T a tableau of shape E with entries ≤ 3, and α, P , Q as above we have This can be seen by applying Remark 2 to the set of representatives X = (1, 3)

The coordinate ring k[Mat m
rs ] is N -graded. Fix a multidegree ν ∈ t . Then one can construct a filtration with sections isomorphic to ∇ GL r (μ) ⊗ ∇ GL s (λ), μ, λ suitable, of the graded piece M 1 of degree ν of k[Mat m rs ] as follows. We use triples (P , Q, α) where P and Q are ordered tableaux of weight ν with shapes λ of length ≤ s and μ of length ≤ r, and α : μ → λ is in a set of (admissible) representatives for the m-tuples of special semi-standard tableaux with shapes determined by Q and weights determined by P . See Section 2.1.

Theorem 1
We can enumerate all the triples (P , Q, α) as above: (P 1 , Q 1 , α 1 ), (P 2 , Q 2 , α 2 ), . . . , (P q , Q q , α q ), λ i the shape of P i , μ i the shape of Q i , such that for all i the span M i of all twisted bideterminants (S | m α j T ), j ≥ i, S of shape μ j with entries ≤ r, T of shape λ j with entries ≤ s, is GL r × GL s -stable and we have an isomorphism Here the λ i are the partitions of t of length ≤ min(mr, s). The isomorphisms to the sections of the filtration are given by After restricting the left multiplication action to GL m r we can decompose the above filtration according to the multidegree in N . From now on we focus on the piece of multidegree ν ∈ t . By repeatedly applying [1,Thm. II.4.11] (see also [11,Thm. 1.4] and Remark 2 after it) to ∇ GL mr (λ i ) we can refine the above filtration to a filtration with sections isomorphic to ⎛ Here the λ i are suitably redefined, the P i go through all ordered tableaux of shape λ i with weight ν, and the Levi GL m r acts on the first factor. The section-isomorphism of [1,Thm. II.4.11] is given by shifting the numbers in each tableau of shape P −1 i (j ) by (j − 1)r, so the result has its entries in (j − 1)r + {1, . . . , r}, and then piecing the resulting tableaux of shapes P −1 i (j ) together according to P i to a tableau of shape λ i . Now we restrict the first factor of (6) to the diagonal copy of GL r in GL m r and we have where for P an ordered tableau with entries ≤ m we put E P = E (P −1 (1),...,P −1 (m)) and for an m-tuple (D 1 , . . . , D m ) of skew Young diagrams where each row or column contains boxes from at most one skew tableau D j . Now we apply [11, Thm. 1.5] and we can refine our previous filtration to a filtration with sections Here the λ i are again suitably redefined and the μ i have length ≤ r. Furthermore, the labelling is coming from triples (P , μ, α) where P is an ordered tableau of weight ν, μ a partition of t and α : μ → E P goes through a set of admissible representatives for the special semi-standard tableaux of shape μ and weight the tuple of row lengths of E P . These triples are in one-one correspondence with the triples (P , Q, α) mentioned earlier.
We now have to check that our filtration is indeed given by spans of twisted bideterminants. From Remark 2 it is clear that under the section-isomorphism (5) the element (S | m α S λ i ) ⊗ (T | S λ i ), S of shape μ with entries ≤ r, α : μ → λ i , T of shape λ i with entries ≤ s, is mapped to (S | m α T ) modulo the (i + 1)-th filtration space. So it now suffices to show that at "stage (7)" the elements (S | m α S λ i ) correspond under the isomorphism (7) combined with the section isomorphism of [1,Thm. II.4.11] to the elements defining the filtration of ∇ GL r (E P i ) from [11,Thm. 1.5].
For this we focus on one particular i which we suppress in the notation. If α : μ → λ is an admissible representative of an m-tuple of special semi-standard tableaux, then the diagram mapping α : μ → E P whose restrictions Q −1 (j ) → P −1 (j ) are the same as those of α, is an admissible representative of the special semi-standard tableau T = S E P • α of shape μ. The elements defining the filtration of ∇ GL r (E P ) from the proof of [11,Thm. 1.5] are (S | α S E P ), S of shape μ with entries ≤ r. Here one should bear in mind that in [11] the bideterminants are formed row-wise rather than column-wise, and that there α −1 is used rather than α: the map f T on page 93 of [11] satisfies (after transposing) T • f T = S E P , and it corresponds to the inverse of our α. 3 By Remark 2 we have where X is a set of representatives for the left cosets of C μ ∩α −1 C E P α in C μ and S α,π (a) = S(π(α −1 (a))) for a ∈ E P . Now we have C μ ∩ α −1 C E P α = C μ (Q) ∩ α −1 C λ (P )α, so, by Remark 2 we have for the same set X where S α,π (a) = S(π(α −1 (a))) + (P (a) − 1)r for a ∈ λ. Under the isomorphism (7) combined with the section isomorphism of [1, Thm. II.4.11] S α,π corresponds to S α,π , that is, (S α,π | S E P ) is mapped to (S α,π | S λ ) modulo the filtration space labelled by "the next P ". So, by the above two equations, (S | α S E P ) is mapped to (S | m α S λ ) modulo the filtration space labelled by the next P . Proof It is easy to see, using Remark 2 for example, that the elements (S μ | m α S λ ) are highest weight vectors of the given weight. Furthermore, they are linearly independent by Theorem 1. On the other hand it follows from standard properties of good filtrations, see [14,Prop. II.4.13], that the dimension of k[Mat m rs ] U r ×U s (μ,λ) is equal to the number of sections . But this is equal to the number of elements of our linearly independent set.
Finally we give a version for the above corollary for the GL r × GL s -action on k[Mat m rs ] defined by ((A, B) , that is, we twist the GL r -action we considered previously with the inverse transpose. We define the anti-canonical tableauS μ of shape μ byS μ (a) = r − a 1 + 1, for a ∈ μ where a 1 is the row index of a. For a tuple μ of integers of length ≤ r we denote by μ rev the reverse of the r-tuple obtained from μ by extending it with zeros. Remark 3 1. We now extract from the proof of Theorem 1 how the triples (P , Q, α) are enumerated. First we order the P 's by identifying each P with the tuple of Young diagrams (i.e. partitions) P −1 ({1, . . . , m − i}) 0≤i≤m−1 and ordering these lexicographically, where the partitions are themselves also ordered lexicographically. For a fixed P we order the pairs (Q, α) as follows. For each i we let S i be the tableau obtained by shifting the entries of S P −1 (i) • α i by i−1 j =0 r j , where r j is the number of rows of P −1 (j ). Here the α i are defined as in Section 2.1. Let S Q,α be the tableau of the same shape as Q obtained by piecing the S i together according to Q. Then we say that (Q 1 , α 1 ) > (Q 2 , α 2 ) if the standard enumeration of S Q 1 ,α 1 is lexicographically less than that of S Q 2 ,α 2 . Now we order the triples (P , Q, α) lexicographically by first comparing the P -component and then the (Q, α)-component. Finally, we enumerate the triples (P , Q, α) in decreasing order. 2. Let E and F be skew Young diagrams with t boxes. In [15,Thm. 3] a basis was given of the space of coinvariants S(E) ⊗ S(F ) Sym t labelled by admissible representatives of special semi-standard tableaux of shape F and weight the tuple of row lengths of E. We give a characteristic free version of this result and two interpretations.
Let r be ≥ the number of rows of F and let s be ≥ the number of rows of E, then the twisted bideterminants (S F | α S E ) ∈ k[Mat rs ] where α goes through a set of admissible representatives of special semi-standard tableaux of shape F and weight the tuple of row lengths of E, are linearly independent.
This can be deduced from [11] as follows. Write F = μ/μ and take E to be E withμ above and to the right of it in such a way that they have no rows or columns in common. We use the definition of Schur modules from Remark 1.1 which uses the right multiplication action. If we combine this with Remark 1.2 we obtain an isomorphism where α : μ → E is given by α| F = α and α|μ = id. For α as above, α goes through a set of representatives for the special tableaux of shape μ and weight the tuple of row lengths of E. Since the elements (S μ | α S E ) are linearly independent by the proof of [11,Thm. 1.5], the result follows. Now we give two interpretations of this result. Firstly, the span of the above bideterminants can be seen as k Secondly, when r ≥ the number of rows of F , this span can be identified with Hom GL r ( GL r (F ), ∇ GL r (E)). Indeed we have by [11, Thm. 1.1(g)] Here we have adapted the notation to that of our paper: We use T E and S E instead of * T E and T E . Furthermore, we associate bideterminants column-wise rather than rowwise, so the bideterminants of shape λ from [11] have shape λ in our notation, U means sum over all tableaux row equivalent to U etc. So the module GL r (F ) is cyclic generated by (S F | T F ) and the homomorphisms are linearly independent, since their images of the generator (S F | T F ) are linearly independent by the above result. We have seen in Section 2.2 that their number is equal to the dimension of Hom GL r ( GL r (F ), ∇ GL r (E)), so they must form a basis. In general the above homomorphisms will always span Hom GL r ( GL r (F ), ∇ GL r (E)) by [9, Prop. 1.5(i)] applied to λ = tε 1 , G = GL r for some r ≥ the number of rows of F and the root system of GL r . For dimension reasons, see Section 2.2, these homomorphisms will then also form a basis when r ≥ the number of rows of E. 3. Assume r = r 1 + · · · + r m for certain integers r j > 0. By similar arguments as in the proof of Theorem 1 one can construct a "good" m j =1 GL r j × GL sfiltration of the degree ν piece of k[Mat rs ] using a spanning set labelled by triples λ, (μ 1 , . . . , μ m ), α , where λ is a partition of t = |ν| of length ≤ s, (μ 1 , . . . , μ m ) is an m-tuple of partitions with μ j of length ≤ r j and |μ 1 | + · · · + |μ m | = t, and where α : E (μ 1 ,...,μ m ) → λ goes through a set of admissible representatives for the special semi-standard tableaux of shape E (μ 1 ,...,μ m ) and weight λ. These triples are in one-one correspondence with the triples (P , (μ 1 , . . . , μ m ), (α 1 , . . . , α m )), where P is an ordered tableau of weight ν, μ 1 , . . . , μ m is an m-tuple of partitions with μ j of length ≤ r j and |μ 1 |+· · ·+|μ m | = t = |ν|, and each α j : μ j → P −1 (j ) goes through a set of admissible representatives for the special semi-standard tableaux of shape μ j and weight the tuple of row lengths of P −1 (j ). The filtration spaces are spanned by twisted bideterminants (S | α T ), where S is of shape E (μ 1 ,...,μ m ) with entries ≤ r, satis- ...,μ m ) for all j , T is of shape λ with entries ≤ s and α : E (μ 1 ,...,μ m ) → λ is as above.

Highest Weight Vectors for the Conjugation Action of GL n on Polynomials
Firstly, let us introduce some further notation. For n a natural number and λ, μ partitions with l(λ) + l(μ) ≤ n, define the descending n-tuple The group GL n acts on Mat n via the conjugation action, given by S · A = SAS −1 and therefore on the coordinate ring k[Mat n ] via (S ·f )(A) = f (S −1 AS) Note that the nilpotent cone N n = {A ∈ Mat n | A n = 0} is under this action a GL n -stable closed subvariety of Mat n . We denote the algebra of invariants of k[Mat n ] under the conjugation action by k[Mat n ] GL n . It is well-known that this is the polynomial algebra in the traces of the exterior powers of the matrix. Now let r, s be integers ≥ 0 with r + s ≤ n. We let GL r × GL s act on k[Mat m rs ] as at the end of Section 3: we use the inverse rather than the transpose to define the action of GL r . For a matrix M denote by M r s the lower left r × s corner of M. For m an integer ≥ 2 we define the map ϕ r,s,n,m : Mat n → Mat m rs by ϕ r,s,n,m (X) = X r s , (X 2 ) r s , . . . , (X m ) r s .
The restriction of this map to the nilpotent cone N n will be denoted by the same symbol. In [15] the following result was proved. Next we recall the following instance of the graded Nakayama Lemma from [15]. Combining Theorem 3 and Lemma 2 we finally obtain Theorem 4 Let χ = [λ, μ] be a dominant weight in the root lattice, l(μ) ≤ r, l(λ) ≤ s, |λ| = |μ| = t, r + s ≤ n. Then the pull-backs of the elements (S μ | m α S λ ), ν, P , Q, α as in Cor. 2 to Thm. 1, along ϕ r,s,n,n−1 : Mat n → Mat n−1 rs span the k[Mat n ] GL n -module k[Mat n ] U n χ .
Remark 4 1. Note that pulling the (S μ | m α S λ ) back just amounts to interpreting x(Q(a)) ij as the (i, j )-th entry of the Q(a)-th matrix power and replacing r −a 1 +1 by n−a 1 +1. In particular, these pulled-back functions don't depend on the choice of r and s. 2. One obtains a bigger, "easier" spanning set by allowing arbitrary P , Q of weight ν and arbitrary bijections α : μ → λ with P • α = Q.
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