Journal of Algebraic Combinatorics

, Volume 39, Issue 2, pp 247–270 | Cite as

Products of Young symmetrizers and ideals in the generic tensor algebra

Article

Abstract

We describe a formula for computing the product of the Young symmetrizer of a Young tableau with the Young symmetrizer of a subtableau, generalizing the classical quasi-idempotence of Young symmetrizers. We derive some consequences to the structure of ideals in the generic tensor algebra and its partial symmetrizations. Instances of these generic algebras appear in the work of Sam and Snowden on twisted commutative algebras, as well as in the work of the author on the defining ideals of secant varieties of Segre–Veronese varieties, and in joint work of Oeding and the author on the defining ideals of tangential varieties of Segre–Veronese varieties.

Keywords

Young symmetrizers Young tableaux Generic tensor algebra 

1 Introduction

In this paper we describe a formula for computing the product of the Young symmetrizer of a Young tableau with the Young symmetrizer of a subtableau. This is a generalization of the classical result which states that an appropriate multiple of a Young symmetrizer is idempotent, and is closely related to the formulas describing the Pieri maps in [8, Sect. 5]. The main motivation for our investigation comes from the study of the equations of special varieties with an action of a product of general linear groups. The GL-modules of equations correspond via Schur–Weyl duality to certain representations of symmetric groups, which we refer to as generic equations. Understanding the ideal structure of the generic equations depends substantially on understanding how the Young symmetrizers multiply. Special instances of our main result (Theorem 1.1) and its application (Theorem 1.5) are implicit in [7, 9] where we establish and generalize conjectures of Garcia–Stillman–Sturmfels and Landsberg–Weyman on the equations of the secant and tangential varieties of a Segre variety (see also Sect. 3.1). We expect that Theorem 1.1 will have applications to a number of other problems such as
  • describing the relations between the minors of a generic matrix [1];

  • proving (weak) Noetherianity of certain twisted commutative algebras generated in degree higher than one [11];

  • determining the equations of secant varieties of Grassmannians.

We introduce some notation before stating our main results: see Sect. 2 for more details, and [2, 3, 5] for more on Young tableaux and the representation theory of symmetric groups. For λ a partition of n (denoted λn), a Young tableau of shape λ is a collection of boxes filled with entries 1,2,…,n, arranged in left-justified rows of lengths λ1λ2≥⋯. For μ a partition with μiλi for all i (denoted μλ), the subtableau S of T of shape μ is obtained by selecting for each i the first μi entries in the ith row of T. For example, for λ=(4,3,1,1) and μ=(2,1,1), one can take We write \(\mathfrak{S}_{n}\) for the symmetric group of permutations of {1,2,…,n} and \(\mathbb{K}[\mathfrak{S}_{n}]\) for its group algebra, where \(\mathbb{K}\) is any field of characteristic zero. With any Young tableau T and subtableau S, as above, we associate the Young symmetrizers\(\mathfrak{c}_{\lambda}(T)\) and \(\mathfrak{c}_{\mu}(S)\), which are elements of \(\mathbb{K}[\mathfrak{S}_{n}]\) (see (2.3) for a precise formula).

Theorem 1.1

Letknbe positive integers, letλn, μkbe partitions withμλ, and letTbe a Young tableau of shapeλcontaining a Young subtableauSof shape μ. We write\(L(T;S)\subset\mathfrak{S}_{n}\)for the set of permutationsσwith the property that for every entrysofS, eitherσ(s)=sorσ(s) lies in a column ofTstrictly to the left of the column ofs, and, moreover, ifσ(s)=sfor allsSthenσ=1is the identity permutation. There exist\(m_{\sigma}\in\mathbb{Q}\)such that
$$ \mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S)= \mathfrak{c}_{\lambda }(T)\cdot\biggl(\sum_{\sigma\in L(T;S)}m_{\sigma} \cdot\sigma\biggr), $$
(1.1)
wherem1=αμis the product of the hook lengths ofμ (see (2.5)). We can takemσ≥0 whenσis an even permutation, andmσ≤0 whenσis odd.

When V is a vector space, λn, and \(\mathfrak {c}_{\lambda}\) is some Young symmetrizer, we can think of multiplication by \(\mathfrak {c}_{\lambda}/\alpha_{\lambda}\) on Vn as a projection VnSλV, where Sλ denotes the Schur functor associated with λ. In particular, (1.1) describes a surjective map SμVV⊗(nk)SλV (equivalently, the expression (1.1) is always non-zero), so it provides some information about the λ-isotypic component of the ideal generated by an irreducible component SμV in the tensor algebra ⨁m≥0Vm. We formulate this more precisely for the generic tensor algebra in Theorem 1.5.

Note that in the case when T=S, formula (1.1) is the classical statement that \(\mathfrak{c}_{\lambda}(T)/\alpha_{\lambda }\) is an idempotent of \(\mathbb{K}[\mathfrak{S}_{n}]\) (see (2.6)). The conclusion that \(m_{\sigma}\in\mathbb{Q}\) is sufficient for our applications, but we believe that a version of (1.1) is valid where mσ are in fact integers. We prove that this is the case when n=k+1 in Theorem 1.2 below, where a precise formula for the coefficients mσ is given. The case n=k+1 of Theorem 1.1 also follows from [8, Theorem 5.2], which however does not guarantee the integrality of the mσ’s. To formulate Theorem 1.2 we need some more notation. We can divide any tableau T of shape λ into rectangular blocks B1,B2,… by grouping together the columns of the same size. For example when λ=(4,3,1,1) there are three blocks B1,B2,B3: The lengths and heights of the blocks Bi are defined in the obvious way: in the example above, they are 1,2,1 and 4,2,1, respectively.

Theorem 1.2

With the notation of Theorem 1.1, assume thatn=k+1 and that the unique entry inToutsideSis equal toa, and is located in position (u,v), i.e. λi=μiforiu, λu=v=μu+1. Let\(\tilde{S}\)be the Young tableau obtained by removing fromSits firstvcolumns: the shape\(\tilde{\mu}\)of\(\tilde{S}\)has\(\tilde{\mu}_{i}=\mu_{i}-v\)fori<u, and\(\tilde{\mu }_{i}=0\)otherwise. Write\(\tilde{B}_{1},\ldots,\tilde{B}_{\tilde{m}}\)for the blocks of\(\tilde{S}\)and denote by\(\tilde{l}_{i}\) (resp. \(\tilde{h}_{i}\)) the length (resp. height) of the block\(\tilde{B}_{i}\). For each\(i=1,\ldots,\tilde{m}\), write\(\tilde{r}_{i}\)for the length of the hook ofμcentered at\((\tilde{h}_{i},v)\), i.e.
$$\tilde{r}_i=\tilde{l}_1+\cdots+\tilde{l}_i+u- \tilde{h}_i. $$
If we define the elements\(\tilde{x}_{i}\in\mathbb{K}[\mathfrak {S}_{n}]\)by\(\tilde{x}_{i}=\sum_{b\in\tilde{B}_{i}} (a,b)\), then
$$ \mathfrak{c}_{\lambda}(T)\cdot\mathfrak {c}_{\mu}(S)= \mathfrak{c}_{\lambda }(T)\cdot\alpha_{\mu}\cdot\prod _{i=1}^{\tilde{m}} \biggl(1-\frac {\tilde{x}_i}{\tilde{r}_i} \biggr). $$
(1.3)

In the case when the blocks \(\tilde{B}_{i}\) consist of a single column (\(\tilde{l}_{i}=1\) for \(i=1,\ldots,\tilde{m}\)), it follows from the discussion following Theorem 1.1 that (1.3) is equivalent to the formulas describing the Pieri maps SμVVSλV in [8, (5.5)], up to a change in the convention used for constructing Young symmetrizers (see also [10]). Our approach offers an alternative to that of [8], in that we work entirely in the group algebra of the symmetric group.

Example 1.3

Let T be as in (1.2), n=9, k=8, and let S be the subtableau of T obtained by removing the box Open image in new window, so (u,v)=(4,1) and a=9. We have \(\tilde{m}=2\), \(\tilde{r}_{1}=4\), \(\tilde{r}_{2}=6\), and
$$\tilde{x}_1=(9,2)+(9,3)+(9,6)+(9,7),\ \tilde{x}_2=(9,4). $$
If we use the same T and take S to be the subtableau obtained by removing Open image in new window, then a=7, (u,v)=(2,3), \(\tilde{m}=1\), \(\tilde{r}_{1}=2\), and \(\tilde{x}_{1}=(7,4)\). Finally, if S is the subtableau obtained by removing Open image in new window then \(\tilde{\mu}\) is empty and (1.3) becomes \(\mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S)= \alpha_{\mu }\cdot\mathfrak{c}_{\lambda}(T)\).

Remark 1.4

When expanding the product formula (1.3), all the denominators are products of distinct hook lengths of the Young diagram μ, so they divide αμ. Moreover, any product of distinct \(\tilde{x}_{i}\)’s is a linear combination of cyclic permutations
$$\sigma=(a,b_1)\cdot(a,b_2)\cdots(a,b_r)=(a,b_r,b_{r-1}, \ldots,b_1), $$
where bi appears to the left of bi+1 in T, in particular σL(T;S). Furthermore, mσ≥0 when r is even, and mσ≤0 when r is odd. We get that Theorem 1.2 implies the special case of Theorem 1.1 when n=k+1.
As an application of Theorem 1.1 we derive certain ideal membership relations of Young symmetrizers with respect to ideals in the generic tensor algebra. We describe a preliminary version of this algebra here in order to state the results, while in Sect. 3 we take a more functorial approach. The generic tensor algebra is the \(\mathbb{K}\)-vector space Open image in new window with multiplication defined on the basis of permutations and extended linearly, as follows. For \(\sigma\in\mathfrak{S}_{n}\) and \(\tau\in\mathfrak{S}_{m}\), \(\sigma *\tau\in\mathfrak{S}_{n+m}\) is given by
$$(\sigma*\tau) (i)= \begin{cases} \sigma(i) & \mbox{for }i\leq n; \\ n + \tau(i-n) & \mbox{for }n+1\leq i\leq n+m. \end{cases} $$
A homogeneous invariant right idealOpen image in new window is a homogeneous right ideal in Open image in new window with the property that each homogeneous component Open image in new window is a left ideal in the group algebra \(\mathbb {K}[\mathfrak{S} _{n}]\). If Open image in new window, we write Open image in new window for the smallest homogeneous invariant right ideal containing A. If S,S′ are Young tableaux with set of entries [k]={1,2,…,k}, we say that SdominatesS if for every i∈[k], writing j (resp. j′) for the column of S (resp. S′) containing i, then jj′. We have the following (see Theorem 3.2 for a stronger statement, and Theorem 3.4 for a partially symmetrized version):

Theorem 1.5

Letknbe positive integers, letλn, μkbe partitions withμλ, and letTbe a Young tableau of shapeλcontaining a Young subtableauSof shapeμand set of entries [k]. Let\(\mathcal{S}\)be the set of Young tableauxSthat have shapeδk, δλ, set of entries [k], and dominateS. We have

Example 1.6

Take n=7, k=5, λ=(4,2,1), μ=(3,2) and The set \(\mathcal{S}\) in Theorem 1.5 consists of the Young tableaux together with the ones obtained from them by permuting the entries within each column.

The structure of the paper is as follows. In Sect. 2 we give some preliminary definitions and results on Young tableaux and Young symmetrizers. In Sect. 3 we describe the generic tensor algebra and a symmetric version of it, together with some consequences of Theorem 1.1 to the ideal structure of these generic algebras. In Sect. 4 we prove Theorem 1.2, and in Sect. 5 we use an inductive argument based on Theorem 1.2 in order to prove Theorem 1.1.

2 Preliminaries

Given a finite set A of size n=|A|, we write \(\mathfrak{S}_{A}\) for the symmetric group of permutations of A. When A={1,…,n}, we simply denote \(\mathfrak{S}_{A}\) by \(\mathfrak{S}_{n}\). If BA then we regard \(\mathfrak{S} _{B}\) as a subgroup of \(\mathfrak{S}_{A}\) in the natural way. We fix a field \(\mathbb{K}\) of characteristic zero, and write \(\mathbb{K}[\mathfrak{S}_{A}]\) for the group algebra of \(\mathfrak{S}_{A}\).

When λn is a partition of n, Dλ={(i,j):1≤jλi} is the associated Young diagram. A Young tableau T of shape λ and set of entries A is a bijection T:DλA. We represent Young diagrams (resp. Young tableaux) pictorially as collections of left-justified rows of boxes (resp. filled boxes) with λi boxes in the ith row, as illustrated in the following example: for A={a,b,c,d,e,f,g} and λ=(4,2,1), we take The (2,1)-entry of T is e. The ith row of T is the set Ri(T)={T(i,j):1≤jλi}, and its jth column Cj(T) is defined analogously. In the example above, C2(T)={a,d}. The conjugate of a partition λ (resp. Young diagram Dλ/tableau T) is obtained by reversing the roles of rows and columns, and is denoted λ′ (resp. Dλ,T′). In our example, T′ has shape λ′=(3,2,1,1). If μ is another partition, we write μλ if μiλi for all i. If μλ then DμDλ and we call the restriction \(S=T|_{D_{\mu}}\) the Young subtableau of T of shape μ. For example, is a Young subtableau of shape μ=(3,1) of the above T.
If X is any subset of A, we write
$$\mathfrak{a}(X)=\sum_{\sigma\in\mathfrak{S}_X} \sigma,\qquad \mathfrak{b}(X)=\sum_{\tau\in\mathfrak{S}_X} \operatorname {sgn}(\tau) \cdot\tau. $$
For \(\delta\in\mathfrak{S}_{A}\) we have \(\mathfrak{a}(\delta (X))=\delta\cdot \mathfrak{a}(X)\cdot\delta^{-1}\), and similarly for \(\mathfrak {b}(\delta(X))\). If \(\delta\in\mathfrak{S}_{X}\) then \(\delta\cdot \mathfrak {a}(X)=\mathfrak{a}(X)\) and \(\delta\cdot\mathfrak {b}(X)=\operatorname{sgn}(\delta)\cdot\mathfrak{b}(X)\), so \(\mathfrak{a}(X)^{2}=|X|!\cdot \mathfrak{a}(X)\) and \(\mathfrak{b}(X)^{2}=|X|!\cdot\mathfrak{b}(X)\). If aAX and if we let z=∑xX(a,x), where \((a,x)\in\mathfrak{S}_{A}\) denotes the transposition of a with x, then
$$ \begin{aligned} \mathfrak{a}\bigl(X\cup \{a\}\bigr)&=\mathfrak{a}(X)\cdot(1+z)=(1+z)\cdot\mathfrak{a}(X), \\ \mathfrak{b}\bigl(X\cup\{a\}\bigr)&=\mathfrak{b}(X)\cdot (1-z)=(1-z)\cdot \mathfrak{b}(X). \end{aligned} $$
(2.1)
If X,YA with |XY|≥2 then \(\mathfrak{a}(X)\cdot \mathfrak{b}(Y)=0\): to see this, consider a transposition \(\tau\in \mathfrak{S} _{X\cap Y}\) and note that
$$\mathfrak{a}(X)\cdot\mathfrak{b}(Y)=\bigl(\mathfrak{a}(X)\cdot \tau\bigr)\cdot \mathfrak{b}(Y)=\mathfrak{a}(X)\cdot\bigl(\tau\cdot\mathfrak {b}(Y)\bigr)= \mathfrak{a}(X)\cdot\bigl(-\mathfrak{b}(Y)\bigr). $$
A similar argument shows that \(\mathfrak{b}(Y)\cdot\mathfrak{a}(X)=0\).
We define the row and column subgroups associated with T as the subgroups of \(\mathfrak{S}_{A}\)
$$ \mathcal{R}_T=\prod _i \mathfrak{S}_{R_i(T)}\quad\mbox{and}\quad \mathcal{C}_T=\prod_j \mathfrak{S}_{C_j(T)}. $$
(2.2)
The Young symmetrizer\(\mathfrak{c}_{\lambda}(T)\) is defined by \(\mathfrak{S}_{A}\) acts naturally on the set of Young tableaux with set of entries A, and we have \(\mathfrak{c}_{\lambda}(\delta\cdot T)=\delta\cdot \mathfrak{c}_{\lambda}\cdot\delta^{-1}\), with similar formulas for \(\mathfrak{a}_{\lambda}(\delta\cdot T)\) and \(\mathfrak{b}_{\lambda }(\delta \cdot T)\).
It follows from (2.1) and the fact that \(\mathfrak{a}(X)\cdot\mathfrak{b}(Y)=0\) when |XY|≥2 that if ij are such that \(\lambda'_{i}\leq\lambda'_{j}\) and aCi(T) then
$$ \mathfrak{c}_{\lambda}(T)\cdot\biggl(1-\sum _{x\in C_j(T)}(a,x) \biggr)=0. $$
(2.4)
This relation is an instance of the Garnir relations [5, Sect. 7].
The hook of λ centered at (x,y) is the subset Hx,y={(x,j):jy}∪{(i,y):ix}⊂Dλ. Its length is the size of Hx,y. We write
$$ \alpha_{\lambda}=\prod_{(i,j)\in D_{\lambda }}|H_{i,j}|. $$
(2.5)
It follows from the Hook Length Formula [2, Sect. 4.1] and [2, Lemma 4.26] that
$$ \mathfrak{c}_{\lambda}(T)^2= \alpha_{\lambda}\cdot\mathfrak{c}_{\lambda}(T). $$
(2.6)

3 Ideals in the generic tensor algebra

In this section we illustrate some applications of Theorem 1.1 to the structure of the ideals in the generic tensor algebra and its partial symmetrizations. Special instances of Theorems 3.2 and 3.4 below were used in [7, 9] in the study of the equations and homogeneous coordinate rings of the secant line and tangential varieties of Segre–Veronese varieties. We illustrate the relevant constructions in the case of the Veronese variety, the extension to the multigraded situation being just a matter of notation.

We write \(\mathit{Vec}_{\mathbb{K}}\) for the category of finite dimensional vector spaces, and Set for the category of finite sets, where morphisms are bijective functions. Note that \(\operatorname {Hom}_{\mathit{Set}}(A,A)=\mathfrak{S} _{A}\), so for any functor Open image in new window, Open image in new window is a \(\mathfrak{S}_{A}\)-representation. By an element of Open image in new window we mean an element of Open image in new window for some ASet. Consider Open image in new window the functor which assigns to a set A with |A|=n, the vector space Open image in new window having a basis consisting of symbols zα where α runs over the set of bijections between [n]={1,…,n} and A. We think of zα as the tensor zα(1)zα(2)⊗⋯⊗zα(n). We have a natural multiplication on Open image in new window, namely for A,BSet we have a map Open image in new window which extends linearly the concatenation of tensors. In terms of the symbols zα, if α:[r]→A and β:[s]→B are bijections, then μA,B(zα,zβ)=zγ, where γ(i)=α(i) for i=1,…,r, and γ(i)=β(ir) for i=r+1,…,r+s. We will often write μA,B(xy) simply as xy. We call Open image in new window the generic tensor algebra. A (right) ideal in Open image in new window is a subfunctor Open image in new window with the property that Open image in new window for all A,BSet. The ideal Open image in new window generated by a set \(\mathcal{E}\) of elements of Open image in new window is the smallest ideal that contains them.

Consider a partition λn and a Young tableau F:Dλ→[n]. We define the Young tabloid associated with F to be the collection Open image in new window of elements of Open image in new window obtained as follows. For any ASet with |A|=n, and any Young tableau T:DλA, we let \(t_{F}(A,T)=\mathfrak{c}_{\lambda}(T)\cdot z_{T\circ F^{-1}}\). We represent a Young tabloid just as a Young tableau, but with the horizontal lines removed: Note that we only construct tabloids from tableaux with entries in [n]. If Fi are Young tableaux of shape λ then a relationiai⋅[Fi]=0 means that for any choice of ASet with |A|=n and of a Young tableau T:DλA, we have Open image in new window. Young tabloids are skew-symmetric in columns, and satisfy the Garnir relations [5, Sect. 7], also known as shuffling relations [13, Sect. 2.1]:

Lemma 3.1

LetF:Dλ→[n] be a Young tableau. The following relations hold:
  1. (a)

    If\(\sigma\in\mathcal{C}_{F}\) (where\(\mathcal{C}_{F}\)is as in (2.2)) then\([\sigma\circ F]=\operatorname {sgn}(\sigma )\cdot[F]\).

     
  2. (b)
    IfXCi(F) andYCi+1(F) with |XY|>|Ci(F)|, then
    $$\sum_{\sigma\in\mathfrak{S}_{X\cup Y}}\operatorname{sgn}(\sigma )\cdot[\sigma \circ F]=0. $$
     
We define the ideal generated by F by Open image in new window, and more generally we can define the ideal generated by a collection of Young tableaux. Observe that in fact Open image in new window for any ASet with |A|=n and any Young tableau T of shape λ and entries in A. To see this, consider another pair (A′,T′) and the corresponding element Open image in new window. Let \(\phi=T'\circ T^{-1}\in \operatorname{Hom}_{\mathit{Set}}(A,A')\). We have
$$t_F\bigl(A',T'\bigr)= \mathfrak{c}_{\lambda}\bigl(T'\bigr)\cdot z_{T'\circ F^{-1}}=\phi \bigl(\mathfrak{c}_{\lambda}(T)\cdot z_{T\circ F^{-1}}\bigr)=\phi \bigl(t_F(A,T)\bigr), $$
so in fact any subfunctor of Open image in new window that contains tF(A,T) will also contain tF(A′,T′), and vice versa.

We write (x1,y1)≺(x2,y2) if y1<y2 and (x1,y1)⪯(x2,y2) if y1y2. If (xi,yi) are the coordinates of a box bi in a Young tableau T, then (x1,y1)≺(x2,y2) means that b1 is contained in a column of T situated to the left of the column of b2.

Theorem 3.2

Letknbe positive integers, letλn, μkbe partitions withμλ, and letF:Dλ→[n] be a Young tableau withF−1([k])=Dμ. Denote by\(\mathcal{G}\)the collection of Young tableauxG:Dλ→[n] with the properties
  1. (1)

    G−1(i)⪯F−1(i) for alli∈[k], withG−1(i)≺F−1(i) for at least onei∈[k].

     
  2. (2)

    G−1([k])=Dδfor some partitionδλ.

     
Writing\(F_{0}=F|_{D_{\mu}}\), we haveIn particular, if we writeG0for the restriction ofGtoG−1([k]) then

Example 3.3

Take n=7, k=5, λ=(4,2,1), μ=(3,2) and Modulo Open image in new window, [F] is a linear combination of tabloids of shape λ containing one of

Proof of Theorem 3.2

Consider any ASet, |A|=n and a Young tableau T:DλA. Let \(T_{0}=T|_{D_{\mu}}\) and A0=T(Dμ). Applying Theorem 1.1 with S=T0 we have
$$ \mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(T_0)= \alpha_{\mu }\cdot\mathfrak{c}_{\lambda}(T)+ \mathfrak{c}_{\lambda}(T) \cdot\biggl(\sum_{\sigma\in L(T;T_0),\sigma\neq\mathbf{1}}m_{\sigma}\cdot\sigma \biggr). $$
(3.3)
Consider the bijection γ:[nk]→(AA0) defined by γ(j)=(TF−1)(j+k), and observe that \(z_{T\circ F^{-1}}=z_{T_{0}\circ F_{0}^{-1}}\cdot z_{\gamma}\). Multiplying both sides of (3.3) by \(z_{T\circ F^{-1}}\) yields To prove (3.1) it is thus enough to show that for σL(T;T0), σ1, \(\mathfrak{c}_{\lambda }(T)\cdot \sigma\cdot z_{T\circ F^{-1}}\) is a linear combination of \(t_{G_{i}}(A,T)\) for \(G_{i}\in\mathcal{G}\). Consider the Young tableaux G=FT−1σ−1T. Since \(\sigma\cdot z_{T\circ F^{-1}}=z_{\sigma\circ T\circ F^{-1}}=z_{T\circ G^{-1}}\), we have \(\mathfrak{c}_{\lambda}(T)\cdot\sigma\cdot z_{T\circ F^{-1}}=t_{G}(A,T)\).
We prove that G satisfies condition (1) of the theorem. To see this, write (x,y)=F−1(i) and let b=T(x,y). We have G−1(i)=T−1(σ(b)), and by the definition of L(T;T0), either σ(b)=b, or σ(b) lies strictly to the left of b in the Young tableau T. It follows that
$$G^{-1}(i)=T^{-1}\bigl(\sigma(b)\bigr)\preceq T^{-1}(b)=(x,y)=F^{-1}(i), $$
with equality if and only if b=σ(b). Since 1 is the only permutation in L(T;T0) that fixes all bA0=(TF−1)([k]), the conclusion follows.

In general it will not be the case that G also satisfies (2), but we can perform a straightening algorithm based on Lemma 3.1 to write [G] as a linear combination of [Gi] with \(G_{i}\in\mathcal{G}\). First of all, using part (a) of Lemma 3.1, we may assume that for each column Ci(G) the entries in [k] appear before those in [nk]. If G−1([k]) is not the Young diagram of a partition, it means that we can find consecutive columns Ci(G), Ci+1(G) such that |Ci(G)∩[k]|<|Ci+1(G)∩[k]|. Let X=Ci(G)∖[k] and Y=Ci+1(G)∩[k]. Applying part (b) of Lemma 3.1, we can write [G] as a linear combination of [Gj] where Ci(G)∩[k]⊆̷Ci(Gj)∩[k]. Condition (1) will be satisfied by the Gj’s since the shuffling relation only moves elements of [k] to the left, so iterating the process yields the desired conclusion.

To prove (3.2), apply (3.1) and induction to each \(G\in\mathcal{G}\) to conclude that Open image in new window. □

3.1 Partially symmetric generic tensor algebras

Starting from the generic tensor algebra Open image in new window, one can construct by partial symmetrization other generic algebras that come up naturally in the study of varieties of tensors with a GL-action. We describe the generic algebras relevant to the study of the secant and tangential variety of a Veronese variety, and leave it to the interested reader to perform the construction in other cases of interest.

Write Setd for the subcategory of Set consisting of sets whose size is divisible by d. Consider the functor Open image in new window which assigns to a set ASetd, |A|=nd, the vector space Open image in new window with basis consisting of monomials \(z_{\underline{A}}=z_{A_{1}}\cdots z_{A_{n}}\) in commuting variables \(z_{A_{i}}\), where \(\underline{A}\) runs over partitions A=A1⊔⋯⊔An with |Ai|=d. In the work of Sam and Snowden [11], Open image in new window is the twisted commutative algebra\(\operatorname {Sym}(\operatorname{Sym}^{d}(\mathbb{C}^{\infty}))\). If we let Open image in new window be the restriction of Open image in new window to Setd, then there is a natural map Open image in new window which is \(\mathbb{K}\)-linear and is defined on the elements of the form zα as follows. For every ASetd, |A|=nd, and bijection α:[nd]→A, we consider the partition \(\underline{A}\) of A obtained by letting Ai=α({d(i−1)+1,…,di}), and define \(\pi (z_{\alpha })=z_{\underline{A}}\). Here π is surjective and multiplicative, so any ideal in Open image in new window is the image of an ideal in Open image in new window. It follows that Theorem 3.2 can be used to derive an analogous result for ideals in Open image in new window, which we explain next.

For a partition λnd, we define a \(\mbox{\textsf{Young}}^{d}_{n}\)tableau of shape λ to be a function F:Dλ→[n] with the property that |F−1(i)|=d for all i∈[n]. If T:DλA is a Young tableau then we write TF−1 for the partition \(\underline{A}\) of A with Ai=T(F−1(i)). We define the \(\mbox{\textsf{Young}}^{d}_{n}\)tabloid associated with F as before, Open image in new window. Note that replacing F by σF for \(\sigma\in\mathfrak{S}_{n}\) permutes the parts of the partition \(\underline{A}\), but preserves \(z_{\underline{A}}\) because of the commutativity of the \(z_{A_{i}}\)’s. It follows that [F]=[σF] for all \(\sigma\in\mathfrak{S}_{n}\). We represent \(\mbox{Young}^{d}_{n}\) tabloids just as the Young tabloids, allowing each entry of [n] to occur exactly d times: for d=3, n=4, λ=(6,3,2,1), a typical \(\mbox{Young}^{3}_{4}\) tabloid of shape λ would be Note that part (a) of Lemma 3.1 implies that if F has repeated entries in some column, then [F]=0. This is the case in the example above.

If X={(xi,yi):i∈[d]}, \(X'=\{(x'_{i},y'_{i}):i\in[d]\}\), with y1y2≤⋯, \(y'_{1}\leq y'_{2}\leq\cdots\), we write XX′ if \(y_{i}\leq y_{i}'\) for all i∈[d], and XX′ if XX′ and \(y_{i}<y'_{i}\) for some i. With this notation, Theorem 3.2 translates almost without change in the partially symmetric setting:

Theorem 3.4

Letknanddbe positive integers, letλnd, μkdbe partitions withμλ, and letF:Dλ→[n] be a\(\mathit{Young}^{d}_{n}\)tableau withF−1([k])=Dμ. Denote by\(\mathcal{G}\)the collection of\(\mathit{Young}^{d}_{n}\)tableauxG:Dλ→[n] with the properties
  1. (1)

    G−1(i)⪯F−1(i) for alli∈[k], withG−1(i)≺F−1(i) for at least onei∈[k].

     
  2. (2)

    G−1([k])=Dδfor some partitionδλ.

     
Writing\(F_{0}=F|_{D_{\mu}}\)we haveIn particular, if we writeG0for the restriction ofGtoG−1([k]) then

The condition \(G\in\mathcal{G}\) can be restated simply by saying that when going from F to G, each entry of F contained in Dμ either remains in the same column, or is moved to the left, the latter situation occurring for at least one such entry. The exact location of an entry within a column is irrelevant due to part (a) of Lemma 3.1.

Proof

Consider a Young tableau \(\tilde{F}:D_{\lambda}\to[nd]\) which is a lifting of the \(\mbox{Young}^{d}_{n}\) tableau F, i.e. \(\tilde{F}\) induces a bijection between F−1(i) and {d(i−1)+1,…,di} for all i∈[n]. By construction we have that \(\pi([\tilde{F}])=[F]\), i.e. \(\pi(t_{\tilde{F}}(A,T))=t_{F}(A,T)\) for all A with |A|=nd and all Young tableaux T:DλA. Letting \(\tilde{F}_{0}=\tilde {F}|_{D_{\mu }}\), we have that \(\pi([\tilde{F}_{0}])=[F_{0}]\). If \(\tilde{G}\) is any Young tableau satisfying conditions (1) and (2) of Theorem 3.2 (with k and n replaced by kd and nd, respectively) then the \(\mbox{Young}^{d}_{n}\) tableau G:Dλ→[n] obtained by \(G=\pi'\circ \tilde{G}\), where π′(j)=i when j∈{d(i−1)+1,…,di}, satisfies conditions (1) and (2) of Theorem 3.4, and, moreover, \(\pi([\tilde{G}])=[G]\). The conclusion of the theorem then follows from that of Theorem 3.2. □

We end with a series of examples of ideals in the generic algebra Open image in new window, explaining their relevance to the study of spaces of tensors. Before that, we introduce one last piece of notation.

For a functor Open image in new window and a partition λnd, write Open image in new window for the subfunctor that assigns to a set ASetd the λ–isotypic component Open image in new window of the \(\mathfrak{S}_{A}\)-representation Open image in new window (note that Open image in new window). A choice of a set A with |A|=nd and of a Young tableau T:DλA, gives rise to a vector space Open image in new window of dimension equal to the multiplicity Open image in new window of the irreducible \(\mathfrak{S}_{A}\)–representation [λ] inside Open image in new window. We call this space a λ-highest weight space of Open image in new window and denote it by Open image in new window. We call the elements of Open image in new windowλ-covariants of Open image in new window. Note that there are choices in the construction of the λ-covariants of Open image in new window, but the subfunctor of Open image in new window that they generate is Open image in new window, which is independent of these choices.

Taking Open image in new window and λnd, the λ-covariants of Open image in new window are just linear combinations of \(\mbox{Young}^{d}_{n}\) tabloids of shape λ. We have that Open image in new window coincides with the multiplicity of the Schur functor Sλ inside the plethysm \(\operatorname{Sym}^{n}\circ\operatorname {Sym}^{d}\). We call Open image in new window the generic version of the polynomial ring \(S=\operatorname{Sym}(\operatorname {Sym}^{d} V)\), which is the homogeneous coordinate ring of the projective space \(\mathbb{P}(\operatorname{Sym}^{d} V)\). More generally, we write \(\mathit{Sch}_{\mathbb{K}}\) for the category of \(\mathbb {K}\)-schemes, and consider a contravariant functor \(X:\mathit{Vec}_{\mathbb{K}}\to \mathit{Sch}_{\mathbb {K}}\), with the property that \(X(V)\subset\mathbb{P}(\operatorname {Sym}^{d} V)\) is a closed subscheme. The ideal of equations I(X(V)) and homogeneous coordinate ring \(\mathbb{K}[X(V)]\) define polynomial functors \(I_{X},S_{X}:\mathit{Vec}_{\mathbb{K}}\to\mathit{Vec}_{\mathbb {K}}\). They have corresponding generic versionsOpen image in new window and Open image in new window defined as follows. For A with |A|=nd, we consider a vector space VA with a basis indexed by the elements of A. The choice of basis on VA gives rise to a maximal torus \(T_{A}\subset\operatorname{GL}(V_{A})\) of diagonal matrices, and there is a natural identification between the (1,1,…,1)-weight space of \(\operatorname{Sym}^{n}(\operatorname{Sym}^{d} V_{A})\) and the vector space Open image in new window. Via this identification, we let Open image in new window be the subspace of Open image in new window that corresponds to the (1,1,…,1)-weight space of \(I_{X}(V)\subset\operatorname{Sym}^{n}(\operatorname{Sym}^{d} V)\). The information encoded by IX,SX is equivalent to that of Open image in new window (see also the discussion on polarization and specialization from [9, Sect. 3C], and [11]).

Example 3.5

(Generic ideals of subspace varieties)

Denote by Open image in new window the ideal generated by the λ-covariants of Open image in new window, where λ runs over partitions with at least k parts. Open image in new window is the generic version of the ideal of a subspace variety [6, Sect. 7.1]: we have Open image in new window, where \(X:\mathit{Vec}_{\mathbb {K}}\to \mathit{Sch}_{\mathbb{K}}\) is defined by letting X(V) be the union of all the subspaces \(\mathbb{P}(\operatorname{Sym}^{d} W)\subset\mathbb{P}(\operatorname{Sym}^{d} V)\), where W runs over the (k−1)-dimensional quotients of V. When k=2, X=Verd is the functor which associates with V the d-th Veronese embedding of \(\mathbb{P}V\).

Example 3.6

(Covariants associated with graphs)

Given a graph Q, we write V(Q) for the set of vertices, and E(Q) for the multiset of edges (we allow multiple edges between two vertices). If Q is an unlabeled graph with n vertices and e edges, with the property that to any vertex there are at most d incident edges, then one constructs a λ-covariant Open image in new window for λ=(nde,e), as follows. Choose a labeling of the vertices of Q with elements of [n] and consider a \(\mbox{Young}^{d}_{n}\) tableau F:Dλ→[n] having a column of size two with entries x,y (in some order) for each edge xyE(Q). The columns of size one of F are such that each element of [n] appears exactly d times in F. As before, [F] denotes the associated \(\mbox{Young}^{d}_{n}\) tabloid. For example, when d=3, r=4 and e=5, typical Q and [F] look like Write [Q] for the tabloid [F]. There are choices in the construction of [Q], but the ideal Open image in new window it generates inside Open image in new window is independent of these choices and is denoted Open image in new window. More generally, if \(\mathcal{Q}\) is a family of graphs, Open image in new window is the ideal generated by the corresponding tabloids.

A direct consequence of Theorem 3.4 is the following.

Proposition 3.7

LetQbe a subgraph of a graphQ. We haveOpen image in new windowwhere\(\mathcal{Q}\)is the set of graphs\(\tilde{Q}\)with\(V(Q')=V(\tilde{Q})\), \(E(Q')\subseteq E(\tilde{Q})\)and\(|E(\tilde{Q})|\leq|E(Q)|\).

Example 3.8

(Generic ideals of secant line varieties [9])

Consider \(\sigma_{2}:\mathit{Vec}_{\mathbb{K}}\to\mathit {Sch}_{\mathbb{K}}\), defined by letting σ2(V) be the variety of secant lines to Verd(V). We have Open image in new window, where Open image in new window is the ideal generated by graphs containing a triangle (i.e. a complete subgraph on 3 vertices). It follows from Proposition 3.7 that Open image in new window is generated in degree three by the graphs on 3 vertices that contain a triangle.

Example 3.9

(Generic ideals of tangential varieties [7])

Consider \(\tau:\mathit{Vec}_{\mathbb{K}}\to\mathit{Sch}_{\mathbb {K}}\), defined by letting τ(V) be the tangential variety to Verd(V). We have Open image in new window, where Open image in new window is the ideal generated by rich graphs, i.e. graphs with more edges than vertices. Open image in new window is generated by graphs with at most 4 vertices. In particular, the ideal of τ is generated in degree at most 4.

4 Proof of Theorem 1.2

We fix a Young tableau S, and the Young tableau T obtained from S by adding one box with entry a in position (u,v). We write μ (resp. λ) for the shape of S (resp. T). We begin by rewriting (1.3) in a more convenient form. We define for 1≤jμ1
$$ z_j=\sum_{b\in C_j(S)}(a,b) \in\mathbb{K}[\mathfrak{S}_n]. $$
(4.1)
Writing μ!=∏iμi!, we note that Using (4.2), (1.3) becomes after multiplying by μ! Consider the blocks B1,…,Bm of the Young tableau obtained by removing the first (v−1) columns of S, denote by li (resp. hi) their lengths (resp. heights), let
$$ x_i=\sum_{b\in B_i}(a,b), $$
(4.4)
and let ri denote the length of the hook of μ centered at (hi,v),
$$ r_i=l_1+\cdots+l_i+h_1-h_i. $$
(4.5)
Note that we have two possibilities:
  1. (1)

    \(\mu'_{v}>\mu'_{v+1}\): in which case \(m=\tilde{m}+1\), B1=Cv(S), \(B_{i}=\tilde{B}_{i-1}\) for i>1.

     
  2. (2)

    \(\mu'_{v}=\mu'_{v+1}\): in which case \(m=\tilde{m}\), \(B_{1}=C_{v}(S)\cup\tilde{B}_{1}\), \(B_{i}=\tilde{B}_{i}\) for i>1.

     

Lemma 4.1

With the notation above, we haveIn particular, (4.3) is equivalent to
$$ \mathfrak{a}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S) \cdot\biggl(1-\frac{x_1}{r_1} \biggr)\cdot\mathfrak{c}_{\mu}(S)= \mathfrak{a}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S)\cdot \alpha_{\mu}\cdot\prod_{i=1}^{m} \biggl(1-\frac{x_i}{r_i} \biggr). $$
(4.8)

We first prove a number of relations that will be useful throughout this section.

Lemma 4.2

If 1≤ijμ1are such that\(\mu'_{i}\leq\mu'_{j}\), then
$$ \begin{aligned} \mathfrak{c}_{\mu}(S)\cdot z_i\cdot z_j &= \mathfrak{c}_{\mu }(S)\cdot z_i, \\ \mathfrak{c}_{\mu}(S)\cdot z_i^2 &= \mathfrak{c}_{\mu}(S)\cdot\bigl(\mu'_i-\bigl( \mu'_i-1\bigr)\cdot z_i\bigr). \end{aligned} $$
(4.9)
As a consequence, for 1≤j<imwe have
$$ \begin{aligned} \mathfrak{c}_{\mu}(S)\cdot x_i\cdot x_j &=\mathfrak{c}_{\mu }(S)\cdot x_i\cdot l_j, \\ \mathfrak{c}_{\mu}(S)\cdot x_i^2 &= \mathfrak{c}_{\mu}(S)\cdot\bigl((l_i-h_i)\cdot x_i + l_i\cdot h_i\bigr). \end{aligned} $$
(4.10)

Proof

We have for 1≤ijμ1 with \(\mu'_{i}\leq\mu'_{j}\) that and (using the fact that \(\mathfrak{c}_{\mu}(S)\cdot(c,b)=-\mathfrak {c}_{\mu}(S)\) if b,c are in the same column of S) To see how (4.10) follows from (4.9), write \(\mathcal{C}_{i}=\{j:C_{j}(S)\subset B_{i}\}\) for the indices of the columns contained in the block Bi. We have Using the fact that \(\mu'_{k}=h_{i}\) if \(k\in\mathcal{C}_{i}\), we get  □

Lemma 4.3

For 1≤j1<j2<⋯<jkμ1, k≥2, and\(b_{i}\in C_{j_{i}}(S)\)a collection of entries lying in distinct columns ofS, we letσbe the cyclic permutation (a,bk,bk−1,…,b1). Ifb1,b2lie in different rows ofSthen\(\mathfrak {c}_{\mu}(S)\cdot\sigma\cdot\mathfrak{c}_{\mu}(S)=0\). Otherwise, lettingτ=(a,bk,bk−1,…,b2), we have\(\mathfrak {c}_{\mu}(S)\cdot\sigma\cdot\mathfrak{c}_{\mu}(S)=\mathfrak {c}_{\mu}(S)\cdot\tau\cdot\mathfrak{c}_{\mu}(S)\). As a consequence,
$$ \mathfrak{c}_{\mu}(S)\cdot\biggl(1-\frac{x_1}{r_1} \biggr)\cdot\mathfrak{c}_{\mu}(S)=\mathfrak{c}_{\mu}(S)\cdot \prod_{i=1}^{m} \biggl(1-\frac{x_i}{r_i} \biggr)\cdot\mathfrak{c}_{\mu}(S). $$
(4.11)

Proof

Suppose first that b1,b2 lie in distinct rows of S, say b2Rs(S), and let x=S(s,j1). Since xb1, σ(x)=x. Since \(\mu'_{j_{1}}!\cdot\mathfrak{c}_{\mu}(S)=\mathfrak{c}_{\mu}(S)\cdot \mathfrak{b}(C_{j_{1}}(S))\) and \(\mu_{s}!\cdot\mathfrak{c}_{\mu }(S)=\mathfrak{a}(R_{s}(S))\cdot\mathfrak{c}_{\mu}(S)\), it is enough to prove that
$$\mathfrak{b}\bigl(C_{j_1}(S)\bigr)\cdot\sigma\cdot\mathfrak{a} \bigl(R_{s}(S)\bigr)=\mathfrak{b}\bigl(C_{j_1}(S)\bigr)\cdot \mathfrak{a}\bigl(\sigma\bigl(R_{s}(S)\bigr)\bigr)\cdot\sigma=0. $$
The last equality holds true because the intersection \(C_{j_{1}}(S)\cap \sigma(R_{s}(S))\) contains at least two elements, namely x=σ(x) and b1=σ(b2).
Suppose now that b1,b2 belong to the same row of S. Since \((b_{1},b_{2})\in\mathcal{R}_{S}\), it follows that \((b_{1},b_{2})\cdot \mathfrak{c}_{\mu}(S)=\mathfrak{c}_{\mu}(S)\). Since σ=τ⋅(b1,b2), we obtain
$$\mathfrak{c}_{\mu}(S)\cdot\sigma\cdot\mathfrak{c}_{\mu }(S)= \mathfrak{c}_{\mu}(S)\cdot\tau\cdot(b_1,b_2)\cdot \mathfrak{c}_{\mu}(S)=\mathfrak{c}_{\mu}(S)\cdot\tau\cdot \mathfrak{c}_{\mu}(S). $$
To prove (4.11) note that it is equivalent, after subtracting the right-hand side from the left and multiplying by r1, to
$$\mathfrak{c}_{\mu}(S)\cdot(r_1-x_1)\cdot \Biggl(1-\prod_{i=2}^{m} \biggl(1- \frac{x_i}{r_i} \biggr) \Biggr)\cdot\mathfrak{c}_{\mu}(S)=0. $$
The left-hand side expands into an expression (with coefficients \(c_{\underline{b}}\in\mathbb{Q}\))
$$\mathfrak{c}_{\mu}(S)\cdot\biggl(r_1-\sum _{b_1\in B_1}(a,b_1) \biggr)\cdot\biggl(\sum _{\substack{2\leq k\leq m\\ 2\leq j_2<\cdots <j_k\leq m\\ b_{i}\in B_{j_i}}} c_{\underline{b}}\cdot(a,b_2)\cdot (a,b_{3})\cdots(a,b_{k}) \biggr)\cdot \mathfrak{c}_{\mu}(S). $$
Fix now 2≤km, 2≤j2<⋯<jkm, and \(b_{i}\in B_{j_{i}}\) for 2≤ik. Letting j1=1, we have by the first part of the lemma that if b1 is not in the same row as b2, and (with the previous notation for τ and σ) if b1 and b2 are in the same row of S. Since there are exactly r1=l1 elements b1B1 lying in the same row as b2, the conclusion follows by summing (4.12) over all such b1’s. □

Proof of Lemma 4.1

Both (4.6) and (4.7) are trivially satisfied when \(\mu'_{v}>\mu'_{v+1}\) because in this case x1=zv, r1=1, \(x_{i}=\tilde{x}_{i-1}\) and \(r_{i}=\tilde{r}_{i-1}\) for i>1. We may then assume that \(\mu'_{v}=\mu'_{v+1}\).

To prove (4.6), since \(\tilde{x}_{i}=x_{i}\) and \(\tilde {r}_{i}=r_{i}\) for i>1, it is enough to show that
$$ \mathfrak{c}_{\mu}(S)\cdot(1-z_v)\cdot \biggl(1-\frac{\tilde {x}_1}{\tilde{r}_1} \biggr)=\mathfrak{c}_{\mu}(S)\cdot\biggl(1- \frac{x_1}{r_1} \biggr). $$
(4.13)
Since \(\tilde{x}_{1}=\sum_{j=v+1}^{v+l_{1}-1} z_{j}\), and \(\mu'_{v}=\mu'_{j}\) for v+1≤jv+l1−1, we get from (4.9) that
$$\mathfrak{c}_{\mu}(S)\cdot z_v\cdot\tilde{x}_1= \mathfrak{c}_{\mu }(S)\cdot z_v\cdot(l_1-1). $$
Using the relation above together with the fact that \(\tilde {r}_{1}=\tilde{l}_{1}+1=l_{1}=r_{1}\), and \(x_{1}=\tilde{x}_{1}+z_{v}\), as desired. To prove (4.7), we multiply (4.13) on the right by \(\mathfrak{c}_{\mu}(S)\), and it remains to show that
$$\mathfrak{c}_{\mu}(S)\cdot(1-z_v)\cdot z_j \cdot\mathfrak{c}_{\mu}(S)=0, $$
for v+1≤jv+l1−1. We have where the second to last equality follows from Lemma 4.3 and the fact that for each b2Cj(S) there exists exactly one b1Cv(S) situated in the same row. □

Proof of Theorem 1.2

Let
$$ P_m=(x_1-r_1) \cdots(x_m-r_m). $$
(4.14)
Using (4.11) we see that (4.8) is equivalent, after multiplying by (−1)mr1rm, to
$$ \mathfrak{a}_{\lambda}(T)\cdot\mathfrak {c}_{\mu}(S) \cdot P_m\cdot\mathfrak{c}_{\mu}(S)=\mathfrak{a}_{\lambda}(T) \cdot\mathfrak{c}_{\mu }(S)\cdot\alpha_{\mu}\cdot P_m. $$
(4.15)
Let
$$ X=\sum_{j=v}^{\mu_1} z_j=\sum_{i=1}^m x_i. $$
(4.16)
We will show in Lemma 4.7 that there exists a polynomial Q(X) (=Qm) with the property
$$ \mathfrak{a}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S) \cdot P_m = \mathfrak{a}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S) \cdot Q(X). $$
(4.17)
Using this, (4.15) is equivalent to
$$\mathfrak{a}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S)\cdot Q(X)\cdot \mathfrak{c}_{\mu}(S)=\mathfrak{a}_{\lambda}(T)\cdot\mathfrak {c}_{\mu }(S)\cdot\alpha_{\mu}\cdot Q(X), $$
which is proved in Lemma 4.6 below. □

Lemma 4.4

Ifzjis as in (4.1) for somej<v, and\(\sigma\in \mathfrak{S} _{n}\)is a permutation that fixes all the entries ofScontained in its first (v−1) columns, then
$$ \mathfrak{a}_{\lambda}(T)\cdot\mathfrak {c}_{\mu}(S) \cdot\sigma\cdot(1-z_j)=0. $$
(4.18)

Proof

If σ(a)≠a, it follows that σ(a) is contained in Ci(S) for some i with \(\mu'_{i}\leq\mu'_{j}\). Since
$$\sigma\cdot(1-z_j)\cdot\sigma^{-1}= \biggl(1-\sum _{x\in C_j(S)}\bigl(\sigma(a),x\bigr) \biggr), $$
(4.18) follows from the Garnir relation (2.4).
We may thus assume that σ(a)=a, so σ⋅(1−zj)⋅σ−1=(1−zj) and we are reduced to the case when σ is the identity. Since \(\mu'_{j}!\cdot\mathfrak{c}_{\mu}(S)=\mathfrak {c}_{\mu}(S)\cdot\mathfrak{b}(C_{j}(S))\) and \(\lambda_{u}!\cdot \mathfrak {a}_{\lambda}(T)=\mathfrak{a}_{\lambda}(T)\cdot\mathfrak {a}(R_{u}(T))\), it suffices to show that
$$\mathfrak{a}\bigl(R_u(T)\bigr)\cdot\mathfrak{c}_{\mu}(S) \cdot\mathfrak{b}\bigl(C_j(S)\bigr)\cdot(1-z_j) \overset{\text{(2.1)}}{=}\mathfrak{a}\bigl(R_u(T) \bigr)\cdot\mathfrak{c}_{\mu }(S)\cdot\mathfrak{b}\bigl(C_j(S) \cup\{a\}\bigr)=0. $$
We show that in fact for any permutation τ appearing in the expansion of \(\mathfrak{c}_{\mu}(S)\), i.e. τ=τ1τ2 with \(\tau_{1}\in\mathcal{R}_{S}\), \(\tau_{2}\in\mathcal{C}_{S}\), we have \(\mathfrak{a}(R_{u}(T))\cdot\tau\cdot\mathfrak{b}(C_{j}(S)\cup\{a\} )=0\), or equivalently
$$\mathfrak{a}\bigl(R_u(T)\bigr)\cdot\tau\cdot\mathfrak{b} \bigl(C_j(S)\cup\{a\} \bigr)\cdot\tau^{-1}=\mathfrak{a} \bigl(R_u(T)\bigr)\cdot\mathfrak{b}\bigl(\tau\bigl(C_j(S) \cup\{a\}\bigr)\bigr)=0. $$
For this it is enough to prove that |Ru(T)∩τ(Cj(S)∪{a})|≥2. Since τ(a)=a and aRu(T), it follows that a is always an element of the intersection. Write b for the (u,j)-entry of S, and observe that since \(\tau_{2}\in\mathcal{C}_{S}\), it induces a permutation of Cj(S), so bτ2(Cj(S)). Since \(\tau_{1}\in\mathcal{R}_{S}\) it follows that τ1(b)∈Ru(S)⊂Ru(T). It follows that the intersection Ru(T)∩τ(Cj(S)∪{a}) contains {a,τ1(b)}, as desired. □

Lemma 4.5

Consider\(\sigma\in\mathfrak{S}_{n}\)withσ(a)=a, and forzjas in (4.1) let
$$ Z=\sum_{j=1}^{\mu_1} z_j=\sum_{b\in S}(a,b). $$
(4.19)
We haveσZ=Zσ, and as a consequence\(\mathfrak{c}_{\mu}(S)\cdot Z=Z\cdot\mathfrak{c}_{\mu}(S)\).

Proof

This relation σZ=Zσ follows from the fact that
$$\sigma\cdot Z\cdot\sigma^{-1}=\sum_{b\in S} \sigma\cdot(a,b)\cdot\sigma^{-1}=\sum_{b\in S} \bigl(a,\sigma(b)\bigr)=\sum_{b\in S}(a,b)=Z. $$
For the consequence, note that all σ in the expansion of \(\mathfrak{c}_{\mu}(S)\) satisfy σ(a)=a. □

Lemma 4.6

Withzjas in (4.1), andXas in (4.16), we have that for any polynomialP(X)
$$ \mathfrak{a}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S) \cdot\alpha_{\mu }\cdot P(X)=\mathfrak{a}_{\lambda}(T)\cdot \mathfrak{c}_{\mu }(S)\cdot P(X)\cdot\mathfrak{c}_{\mu}(S). $$
(4.20)

Proof

By linearity, it suffices to prove (4.20) when P(X)=Xt. We argue by induction on t: for t=0, the result follows from (2.6). Assume now that t>0. Expanding Xt−1, we get a linear combination of permutations σ that fix all the entries in the first (v−1) columns of S. Using Lemma 4.4 and letting Z as in (4.19), we get Multiplying both sides by αμ and using the induction hypothesis and the fact that Z and \(\mathfrak{c}_{\mu}(S)\) commute (Lemma 4.5), we get Applying (4.21) again to the last term of the equality yields the desired conclusion. □
The last ingredient of the proof of Theorem 1.2 is (4.17), which we prove in the rest of this section. We define for t=1,…,m (recall the definition of li,hi and ri from (4.5)) Note that Xm=X (as defined in (4.16)), that Qt is a polynomial in Xt, and that the definition of Pm coincides with that in (4.14). For an element \(f\in\mathbb{K}[\mathfrak{S}_{n}]\), we define its right annihilator by
$$\mathit{RAnn}(f)=\bigl\{p\in\mathbb{K}[\mathfrak{S}_n]:f\cdot p=0 \bigr\}. $$
We define a congruence relation ≡ on \(\mathbb{K}[\mathfrak {S}_{n}]\) by Since I is a right ideal, fg implies fhgh, but in general \(hf\not\equiv hg\). However, for any polynomial P(X) (where X is as defined in (4.16)) we have
$$ f\equiv g\quad\Longrightarrow\quad P(X)\cdot f\equiv P(X) \cdot g. $$
(4.24)
Moreover, we have
$$ c_{\mu}(S)\cdot f=c_{\mu}(S)\cdot g\quad \Longrightarrow\quad f\equiv g. $$
(4.25)
This is because (4.23) is equivalent after multiplying both sides by αμ and using (4.20) to
$$f\equiv g\quad\Longleftrightarrow\quad\mathfrak{a}_{\lambda }(T)\cdot c_{\mu }(S)\cdot X^i\cdot c_{\mu}(S)\cdot f= \mathfrak{a}_{\lambda}(T)\cdot c_{\mu}(S)\cdot X^i\cdot c_{\mu}(S)\cdot g\quad\forall i\geq0. $$
It follows from (4.10) and (4.25) that
$$ \begin{aligned} x_i\cdot x_j &\equiv x_i\cdot l_j, \\ x_i^2 &\equiv(l_i-h_i)\cdot x_i + l_i\cdot h_i. \end{aligned} $$
(4.26)

We will show that PmQ(X) for some polynomial Q (namely Q(X)=Qm), which will imply (4.17) and thus conclude the proof of Theorem 1.2. We will prove by induction on t the following

Lemma 4.7

Withas defined in (4.23), the following relations hold for 1≤tm:
(at)

PtQt.

(bt)

Pt⋅(Xt+h1)≡(Xt+h1)⋅Pt≡0.

Before that, we formulate some preliminary results.

Lemma 4.8

For 2≤jmandt≥1 we have
$$ \begin{aligned} \mathfrak{c}_{\mu}(S)\cdot x_j\cdot P_t &= 0, \\ \mathfrak{c}_{\mu}(S)\cdot(x_1+h_1)\cdot P_t &= 0. \end{aligned} $$
(4.27)
IfQ(Xt) is any polynomial inXtandt+1≤jm, then
$$ \mathfrak{c}_{\mu}(S)\cdot x_j\cdot Q(X_t)= \mathfrak{c}_{\mu }(S)\cdot x_j\cdot Q \bigl(s_t^t\bigr). $$
(4.28)
In particular, fort+1≤jmwe have
$$ \mathfrak{c}_{\mu}(S)\cdot x_j\cdot Q_t=0. $$
(4.29)

Proof

Relations (4.27) follows from \(\mathfrak{c}_{\mu}(S)\cdot x_{j}\cdot(x_{1}-l_{1})=0\) for j≥2, and \(\mathfrak{c}_{\mu}(S)\cdot (x_{1}+h_{1})\cdot(x_{1}-l_{1})=0\), which are special cases of (4.10).

Since j>t, we have by (4.10) that
$$\mathfrak{c}_{\mu}(S)\cdot x_j\cdot X_t = \mathfrak{c}_{\mu }(S)\cdot x_j\cdot(x_1+ \cdots+x_t) = \mathfrak{c}_{\mu}(S)\cdot x_j \cdot(l_1+\cdots+l_t) = \mathfrak{c}_{\mu}(S) \cdot x_j\cdot s_t^t, $$
which when applied iteratively yields (4.28). Relation (4.29) now follows from (4.28) and the fact that Qt is a polynomial in Xt divisible by \((X_{t}-s_{t}^{t})\) (see (4.22)). □

Lemma 4.9

For 2≤jmandt≥1 we have
$$ \begin{aligned} x_j\cdot P_t &\equiv0, \\ (x_1+h_1)\cdot P_t &\equiv0. \end{aligned} $$
(4.30)
IfQ(Xt) is any polynomial inXtandt+1≤jm, then
$$ x_j\cdot Q(X_t) \equiv x_j\cdot Q\bigl(s_t^t\bigr). $$
(4.31)
In particular, fort+1≤jmwe have
$$ x_j\cdot Q_t\equiv0. $$
(4.32)

Proof

This follows from Lemma 4.8 using (4.25). □

Lemma 4.10

IfQis a polynomial then
$$ Q(X_t)\cdot\bigl(X_t-s_t^t \bigr)\equiv Q(X)\cdot\bigl(X_t-s_t^t\bigr). $$
(4.33)

Proof

We prove by induction on i that \(X^{i}\cdot(X_{t}-s_{t}^{t})\equiv X_{t}^{i}\cdot(X_{t}-s_{t}^{t})\), the result being trivial for i=0. Assuming that the result is true for some i and multiplying both sides by X, we get using (4.24) that  □

Proof of Lemma 4.7

We do induction on t, showing that (at)⇒(bt) and (at),(bt)⇒(at+1). When t=1, we have \(r_{1}=s_{1}^{1}=l_{1}\) so P1=Q1=x1l1 and (a1) holds. We have \(Q_{t}=Q(X_{t})\cdot(X_{t}-s_{t}^{t})\), where Q is the polynomial \(Q(z)=\prod_{i=1}^{t-1}(z-s^{t}_{i})\). By (4.33), \(Q_{t}\equiv Q(X)\cdot(X_{t}-s_{t}^{t})\), so (4.34) becomes
$$ P_{t+1}\equiv Q(X)\cdot\bigl(X_t-s_t^t \bigr)\cdot\bigl(X_{t+1}+h_{t+1}-s_{t+1}^{t+1} \bigr). $$
(4.35)
We have By (4.24) this chain of congruences is preserved if we multiply on the left by Q(X), so (4.35) is equivalent to
$$ P_{t+1}\equiv Q(X)\cdot\bigl(X-s^{t+1}_t \bigr)\cdot\bigl(X_{t+1}-s^{t+1}_{t+1}\bigr). $$
(4.36)
Using (4.33) with t replaced by (t+1) and Q(X) replaced by \(Q(X)\cdot(X-s^{t+1}_{t})\), and using the fact that \(s_{i}^{t}=s_{i}^{t+1}\) for it−1, we obtain
$$P_{t+1}\equiv Q(X_{t+1})\cdot\bigl(X_{t+1}-s^{t+1}_t \bigr)\cdot\bigl(X_{t+1}-s^{t+1}_{t+1} \bigr)=Q_{t+1}, $$
which concludes the proof of Lemma 4.7 and that of Theorem 1.2. □

5 Proof of Theorem 1.1

We prove Theorem 1.1 by induction on the difference (nk). When nk=0, the theorem follows from the quasi-idempotence of Young symmetrizers (2.6). When nk=1, the theorem is a consequence of Theorem 1.2 (Remark 1.4). We may thus assume that nk≥2.

We consider the subtableau U of T obtained by removing the rightmost corner box of T not contained in S. More precisely, U is obtained by removing the box in position (u,v) where \(v=\max\{j:\lambda'_{j}\neq\mu'_{j}\}\) and \(u=\lambda'_{v}\). We write δ for the shape of U. For example, take n=8, k=4, λ=(3,2,2,1), μ=(3,1), in which case (u,v)=(3,2) and δ=(3,2,1,1):
We have by induction
$$ \mathfrak{c}_{\delta}(U)\cdot\mathfrak{c}_{\mu}(S)= \mathfrak{c}_{\delta}(U)\cdot\biggl(\sum_{\sigma\in L(U;S)}m_{\sigma} \cdot\sigma\biggr) $$
(5.1)
with m1=αμ. By Theorem 1.2
$$ \mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\delta}(U) = \mathfrak{c}_{\lambda}(T)\cdot\biggl(\sum_{\tau\in L(T;U)}n_{\tau} \cdot\tau\biggr) $$
(5.2)
where n1=αδ, and, moreover, nτ≠0 only for permutations τ that fix the entries in the first v columns of U. By the choice of (u,v), all entries of U in the columns v+1,v+2,… belong to S, so the permutations τ appearing in (5.2) fix all the entries of U outside S. Combining (5.1) with (5.2) we obtain
$$ \mathfrak{c}_{\lambda}(T)\cdot\mathfrak {c}_{\delta}(U) \cdot\mathfrak{c}_{\mu}(S)=\mathfrak{c}_{\lambda}(T)\cdot\biggl( \sum_{\tau\in L(T;U)}n_{\tau}\cdot\tau\biggr)\cdot \biggl(\sum_{\sigma\in L(U;S)}m_{\sigma}\cdot\sigma\biggr). $$
(5.3)
It is clear that if τL(T;U) and σL(U;S), then for every sS either (τσ)(s)=s or (τσ)(s) lies strictly to the left of s. Assume now that (τσ)(s)=s for all entries s in S. Since every τ appearing with non-zero coefficient in (5.2) fixes the entries of U outside S, it must be that σ permutes the entries of S, but then the definition of L(U;S) yields σ=1. It follows that τ fixes the entries of S along with those of US, so τ=1.
We claim that
$$ \mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(S)= \frac {1}{\alpha _{\delta}}\cdot\mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\delta }(U) \cdot\mathfrak{c}_{\mu}(S). $$
(5.4)
To prove (5.4) it suffices to show that for every τ1 appearing in (5.2) with non-zero coefficient, we have \(\mathfrak{c}_{\lambda}(T)\cdot\tau\cdot \mathfrak {c}_{\mu}(S)=0\), or equivalently (after multiplying on the right by τ−1) \(\mathfrak{c}_{\lambda}(T)\cdot\mathfrak{c}_{\mu }(\tau \cdot S)=0\). Since τ1, there exists an entry s in S such that τ(s)=a is the unique entry of T outside U (s=b1 in the notation of Remark 1.4). If we write (i,j) for the coordinates of s, we have j>v. Let s′=S(i,v) and note that τ(s′)=s′. We have that a and s′ are both contained in Cv(T) and in Ri(τS), which forces \(\mathfrak {c}_{\lambda}(T)\cdot\mathfrak{c}_{\mu}(\tau\cdot S)=0\), as desired. The conclusion of Theorem 1.1 now follows by expanding (5.3) and using (5.4).

Notes

Acknowledgements

I would like to thank Steven Sam and John Stembridge for kindly answering my questions about the project, and the anonymous referees for many helpful suggestions. Experiments with the computer algebra softwares Macaulay 2 [4] and Sage [12] have provided numerous valuable insights.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Romanian AcademyInstitute of Mathematics “Simion Stoilow”BucharestRomania

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