# Products of Young symmetrizers and ideals in the generic tensor algebra

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## 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

*λ*a partition of

*n*(denoted

*λ*⊢

*n*), a Young tableau of shape

*λ*is a collection of boxes filled with entries 1,2,…,

*n*, arranged in left-justified rows of lengths

*λ*

_{1}≥

*λ*

_{2}≥⋯. For

*μ*a partition with

*μ*

_{ i }≤

*λ*

_{ i }for all

*i*(denoted

*μ*⊂

*λ*), the subtableau

*S*of

*T*of shape

*μ*is obtained by selecting for each

*i*the first

*μ*

_{ i }entries in the

*i*th 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

*Let*

*k*≤

*n*

*be positive integers*,

*let*

*λ*⊢

*n*,

*μ*⊢

*k*

*be partitions with*

*μ*⊂

*λ*,

*and let*

*T*

*be a Young tableau of shape*

*λ*

*containing a Young subtableau*

*S*

*of shape μ*.

*We write*\(L(T;S)\subset\mathfrak{S}_{n}\)

*for the set of permutations*

*σ*

*with the property that for every entry*

*s*

*of*

*S*,

*either*

*σ*(

*s*)=

*s*

*or*

*σ*(

*s*)

*lies in a column of*

*T*

*strictly to the left of the column of*

*s*,

*and*,

*moreover*,

*if*

*σ*(

*s*)=

*s*

*for all*

*s*∈

*S*

*then*

*σ*=

**1**

*is the identity permutation*.

*There exist*\(m_{\sigma}\in\mathbb{Q}\)

*such that*

*where*

*m*

_{ 1 }=

*α*

_{ μ }

*is the product of the hook lengths of*

*μ*(

*see*(2.5)).

*We can take*

*m*

_{ σ }≥0

*when*

*σ*

*is an even permutation*,

*and*

*m*

_{ σ }≤0

*when*

*σ*

*is odd*.

When *V* is a vector space, *λ*⊢*n*, and \(\mathfrak {c}_{\lambda}\) is some Young symmetrizer, we can think of multiplication by \(\mathfrak {c}_{\lambda}/\alpha_{\lambda}\) on *V* ^{⊗n } as a projection *V* ^{⊗n }→*S* _{ λ } *V*, where *S* _{ λ } denotes the Schur functor associated with *λ*. In particular, (1.1) describes a surjective map *S* _{ μ } *V*⊗*V* ^{⊗(n−k)}→*S* _{ λ } *V* (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 *S* _{ μ } *V* in the tensor algebra ⨁_{ m≥0} *V* ^{⊗m }. We formulate this more precisely for the generic tensor algebra in Theorem 1.5.

*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

*m*

_{ σ }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

*m*

_{ σ }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

*m*

_{ σ }’s. To formulate Theorem 1.2 we need some more notation. We can divide any tableau

*T*of shape

*λ*into rectangular blocks

*B*

_{1},

*B*

_{2},… by grouping together the columns of the same size. For example when

*λ*=(4,3,1,1) there are three blocks

*B*

_{1},

*B*

_{2},

*B*

_{3}:

*B*

_{ i }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 that*

*n*=

*k*+1

*and that the unique entry in*

*T*

*outside*

*S*

*is equal to*

*a*,

*and is located in position*(

*u*,

*v*),

*i*.

*e*.

*λ*

_{ i }=

*μ*

_{ i }

*for*

*i*≠

*u*,

*λ*

_{ u }=

*v*=

*μ*

_{ u }+1.

*Let*\(\tilde{S}\)

*be the Young tableau obtained by removing from*

*S*

*its first*

*v*

*columns*:

*the shape*\(\tilde{\mu}\)

*of*\(\tilde{S}\)

*has*\(\tilde{\mu}_{i}=\mu_{i}-v\)

*for*

*i*<

*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 of*

*μ*

*centered at*\((\tilde{h}_{i},v)\),

*i*.

*e*.

*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*

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 *S* _{ μ } *V*⊗*V*→*S* _{ λ } *V* 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

*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

*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

*μ*, so they divide

*α*

_{ μ }. Moreover, any product of distinct \(\tilde{x}_{i}\)’s is a linear combination of cyclic permutations

*b*

_{ i }appears to the left of

*b*

_{ i+1}in

*T*, in particular

*σ*∈

*L*(

*T*;

*S*). Furthermore,

*m*

_{ σ }≥0 when

*r*is even, and

*m*

_{ σ }≤0 when

*r*is odd. We get that Theorem 1.2 implies the special case of Theorem 1.1 when

*n*=

*k*+1.

*A*. If

*S*,

*S*′ are Young tableaux with set of entries [

*k*]={1,2,…,

*k*}, we say that

*S*′ dominates

*S*if for every

*i*∈[

*k*], writing

*j*(resp.

*j*′) for the column of

*S*(resp.

*S*′) containing

*i*, then

*j*≥

*j*′. We have the following (see Theorem 3.2 for a stronger statement, and Theorem 3.4 for a partially symmetrized version):

## Theorem 1.5

*Let*

*k*≤

*n*

*be positive integers*,

*let*

*λ*⊢

*n*,

*μ*⊢

*k*

*be partitions with*

*μ*⊂

*λ*,

*and let*

*T*

*be a Young tableau of shape*

*λ*

*containing a Young subtableau*

*S*

*of shape*

*μ*

*and set of entries*[

*k*].

*Let*\(\mathcal{S}\)

*be the set of Young tableaux*

*S*′

*that have shape*

*δ*⊢

*k*,

*δ*⊂

*λ*,

*set of entries*[

*k*],

*and dominate*

*S*.

*We have*

## Example 1.6

*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 *B*⊂*A* 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 \(\mathfrak{S}_{A}\).

*λ*⊢

*n*is a partition of

*n*,

*D*

_{ λ }={(

*i*,

*j*):1≤

*j*≤

*λ*

_{ i }} is the associated Young diagram. A Young tableau

*T*of shape

*λ*and set of entries

*A*is a bijection

*T*:

*D*

_{ λ }→

*A*. We represent Young diagrams (resp. Young tableaux) pictorially as collections of left-justified rows of boxes (resp. filled boxes) with

*λ*

_{ i }boxes in the

*i*th 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

*i*th row of

*T*is the set

*R*

_{ i }(

*T*)={

*T*(

*i*,

*j*):1≤

*j*≤

*λ*

_{ i }}, and its

*j*th column

*C*

_{ j }(

*T*) is defined analogously. In the example above,

*C*

_{2}(

*T*)={

*a*,

*d*}. The conjugate of a partition

*λ*(resp. Young diagram

*D*

_{ λ }/tableau

*T*) is obtained by reversing the roles of rows and columns, and is denoted

*λ*′ (resp.

*D*

_{ λ′},

*T*′). In our example,

*T*′ has shape

*λ*′=(3,2,1,1). If

*μ*is another partition, we write

*μ*⊂

*λ*if

*μ*

_{ i }≤

*λ*

_{ i }for all

*i*. If

*μ*⊂

*λ*then

*D*

_{ μ }⊂

*D*

_{ λ }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*.

*X*is any subset of

*A*, we write

*a*∈

*A*∖

*X*and if we let

*z*=∑

_{ x∈X }(

*a*,

*x*), where \((a,x)\in\mathfrak{S}_{A}\) denotes the transposition of

*a*with

*x*, then

*X*,

*Y*⊂

*A*with |

*X*∩

*Y*|≥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

*T*as the subgroups of \(\mathfrak{S}_{A}\)

*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 \(\mathfrak{b}_{\lambda }(\delta \cdot T)\).

*X*∩

*Y*|≥2 that if

*i*≠

*j*are such that \(\lambda'_{i}\leq\lambda'_{j}\) and

*a*∈

*C*

_{ i }(

*T*) then

*λ*centered at (

*x*,

*y*) is the subset

*H*

_{ x,y }={(

*x*,

*j*):

*j*≥

*y*}∪{(

*i*,

*y*):

*i*≥

*x*}⊂

*D*

_{ λ }. Its length is the size of

*H*

_{ x,y }. We write

## 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 *A*∈*Set*. 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 *z* _{ α } where *α* runs over the set of bijections between [*n*]={1,…,*n*} and *A*. We think of *z* _{ α } as the tensor *z* _{ α(1)}⊗*z* _{ α(2)}⊗⋯⊗*z* _{ α(n)}. We have a natural multiplication on Open image in new window , namely for *A*,*B*∈*Set* we have a map Open image in new window which extends linearly the concatenation of tensors. In terms of the symbols *z* _{ α }, if *α*:[*r*]→*A* and *β*:[*s*]→*B* are bijections, then *μ* _{ A,B }(*z* _{ α },*z* _{ β })=*z* _{ γ }, where *γ*(*i*)=*α*(*i*) for *i*=1,…,*r*, and *γ*(*i*)=*β*(*i*−*r*) for *i*=*r*+1,…,*r*+*s*. We will often write *μ* _{ A,B }(*x*⊗*y*) simply as *x*⋅*y*. 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*,*B*∈*Set*. 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.

*λ*⊢

*n*and a Young tableau

*F*:

*D*

_{ λ }→[

*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

*A*∈

*Set*with |

*A*|=

*n*, and any Young tableau

*T*:

*D*

_{ λ }→

*A*, 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

*F*

_{ i }are Young tableaux of shape

*λ*then a relation ∑

_{ i }

*a*

_{ i }⋅[

*F*

_{ i }]=0 means that for any choice of

*A*∈

*Set*with |

*A*|=

*n*and of a Young tableau

*T*:

*D*

_{ λ }→

*A*, 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

*Let*

*F*:

*D*

_{ λ }→[

*n*]

*be a Young tableau*.

*The following relations hold*:

- (a)
*If*\(\sigma\in\mathcal{C}_{F}\) (*where*\(\mathcal{C}_{F}\)*is as in*(2.2))*then*\([\sigma\circ F]=\operatorname {sgn}(\sigma )\cdot[F]\). - (b)
*If**X*⊂*C*_{ i }(*F*)*and**Y*⊂*C*_{ i+1}(*F*)*with*|*X*∪*Y*|>|*C*_{ i }(*F*)|,*then*$$\sum_{\sigma\in\mathfrak{S}_{X\cup Y}}\operatorname{sgn}(\sigma )\cdot[\sigma \circ F]=0. $$

*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

*A*∈

*Set*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 }(

*A*,

*T*) will also contain

*t*

_{ F }(

*A*′,

*T*′), and vice versa.

We write (*x* _{1},*y* _{1})≺(*x* _{2},*y* _{2}) if *y* _{1}<*y* _{2} and (*x* _{1},*y* _{1})⪯(*x* _{2},*y* _{2}) if *y* _{1}≤*y* _{2}. If (*x* _{ i },*y* _{ i }) are the coordinates of a box *b* _{ i } in a Young tableau *T*, then (*x* _{1},*y* _{1})≺(*x* _{2},*y* _{2}) means that *b* _{1} is contained in a column of *T* situated to the left of the column of *b* _{2}.

## Theorem 3.2

*Let*

*k*≤

*n*

*be positive integers*,

*let*

*λ*⊢

*n*,

*μ*⊢

*k*

*be partitions with*

*μ*⊂

*λ*,

*and let*

*F*:

*D*

_{ λ }→[

*n*]

*be a Young tableau with*

*F*

^{−1}([

*k*])=

*D*

_{ μ }.

*Denote by*\(\mathcal{G}\)

*the collection of Young tableaux*

*G*:

*D*

_{ λ }→[

*n*]

*with the properties*

- (1)
*G*^{−1}(*i*)⪯*F*^{−1}(*i*)*for all**i*∈[*k*],*with**G*^{−1}(*i*)≺*F*^{−1}(*i*)*for at least one**i*∈[*k*]. - (2)
*G*^{−1}([*k*])=*D*_{ δ }*for some partition**δ*⊂*λ*.

*Writing*\(F_{0}=F|_{D_{\mu}}\),

*we have*

*In particular*,

*if we write*

*G*

_{0}

*for the restriction of*

*G*

*to*

*G*

^{−1}([

*k*])

*then*

## Example 3.3

*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

*A*∈

*Set*, |

*A*|=

*n*and a Young tableau

*T*:

*D*

_{ λ }→

*A*. Let \(T_{0}=T|_{D_{\mu}}\) and

*A*

_{0}=

*T*(

*D*

_{ μ }). Applying Theorem 1.1 with

*S*=

*T*

_{0}we have

*γ*:[

*n*−

*k*]→(

*A*∖

*A*

_{0}) defined by

*γ*(

*j*)=(

*T*∘

*F*

^{−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

*σ*∈

*L*(

*T*;

*T*

_{0}),

*σ*≠

**1**, \(\mathfrak{c}_{\lambda }(T)\cdot \sigma\cdot z_{T\circ F^{-1}}\) is a linear combination of \(t_{G_{i}}(A,T)\) for \(G_{i}\in\mathcal{G}\). Consider the Young tableaux

*G*=

*F*∘

*T*

^{−1}∘

*σ*

^{−1}∘

*T*. Since \(\sigma\cdot z_{T\circ F^{-1}}=z_{\sigma\circ T\circ F^{-1}}=z_{T\circ G^{-1}}\), we have \(\mathfrak{c}_{\lambda}(T)\cdot\sigma\cdot z_{T\circ F^{-1}}=t_{G}(A,T)\).

*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*;

*T*

_{0}), either

*σ*(

*b*)=

*b*, or

*σ*(

*b*) lies strictly to the left of

*b*in the Young tableau

*T*. It follows that

*b*=

*σ*(

*b*). Since

**1**is the only permutation in

*L*(

*T*;

*T*

_{0}) that fixes all

*b*∈

*A*

_{0}=(

*T*∘

*F*

^{−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 [*G* _{ i }] with \(G_{i}\in\mathcal{G}\). First of all, using part (a) of Lemma 3.1, we may assume that for each column *C* _{ i }(*G*) the entries in [*k*] appear before those in [*n*−*k*]. If *G* ^{−1}([*k*]) is not the Young diagram of a partition, it means that we can find consecutive columns *C* _{ i }(*G*), *C* _{ i+1}(*G*) such that |*C* _{ i }(*G*)∩[*k*]|<|*C* _{ i+1}(*G*)∩[*k*]|. Let *X*=*C* _{ i }(*G*)∖[*k*] and *Y*=*C* _{ i+1}(*G*)∩[*k*]. Applying part (b) of Lemma 3.1, we can write [*G*] as a linear combination of [*G* _{ j }] where *C* _{ i }(*G*)∩[*k*]⊆̷*C* _{ i }(*G* _{ j })∩[*k*]. Condition (1) will be satisfied by the *G* _{ j }’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 *Set* ^{ d } 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 *A*∈*Set* ^{ d }, |*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*=*A* _{1}⊔⋯⊔*A* _{ n } with |*A* _{ i }|=*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 *Set* ^{ d }, 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 *z* _{ α } as follows. For every *A*∈*Set* ^{ d }, |*A*|=*nd*, and bijection *α*:[*nd*]→*A*, we consider the partition \(\underline{A}\) of *A* obtained by letting *A* _{ i }=*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.

*λ*⊢

*nd*, we define a \(\mbox{\textsf{Young}}^{d}_{n}\) tableau of shape

*λ*to be a function

*F*:

*D*

_{ λ }→[

*n*] with the property that |

*F*

^{−1}(

*i*)|=

*d*for all

*i*∈[

*n*]. If

*T*:

*D*

_{ λ }→

*A*is a Young tableau then we write

*T*∘

*F*

^{−1}for the partition \(\underline{A}\) of

*A*with

*A*

_{ i }=

*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

*σ*∘

*F*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*]=[

*σ*∘

*F*] 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*={(*x* _{ i },*y* _{ i }):*i*∈[*d*]}, \(X'=\{(x'_{i},y'_{i}):i\in[d]\}\), with *y* _{1}≤*y* _{2}≤⋯, \(y'_{1}\leq y'_{2}\leq\cdots\), we write *X*⪯*X*′ if \(y_{i}\leq y_{i}'\) for all *i*∈[*d*], and *X*≺*X*′ if *X*⪯*X*′ 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

*Let*

*k*≤

*n*

*and*

*d*

*be positive integers*,

*let*

*λ*⊢

*nd*,

*μ*⊢

*kd*

*be partitions with*

*μ*⊂

*λ*,

*and let*

*F*:

*D*

_{ λ }→[

*n*]

*be a*\(\mathit{Young}^{d}_{n}\)

*tableau with*

*F*

^{−1}([

*k*])=

*D*

_{ μ }.

*Denote by*\(\mathcal{G}\)

*the collection of*\(\mathit{Young}^{d}_{n}\)

*tableaux*

*G*:

*D*

_{ λ }→[

*n*]

*with the properties*

- (1)
*G*^{−1}(*i*)⪯*F*^{−1}(*i*)*for all**i*∈[*k*],*with**G*^{−1}(*i*)≺*F*^{−1}(*i*)*for at least one**i*∈[*k*]. - (2)
*G*^{−1}([*k*])=*D*_{ δ }*for some partition**δ*⊂*λ*.

*Writing*\(F_{0}=F|_{D_{\mu}}\)

*we have*

*In particular*,

*if we write*

*G*

_{0}

*for the restriction of*

*G*

*to*

*G*

^{−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 *D* _{ μ } 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*:*D* _{ λ }→*A*. 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*:*D* _{ λ }→[*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 *λ*⊢*nd*, write Open image in new window for the subfunctor that assigns to a set *A*∈*Set* ^{ d } 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*:*D* _{ λ }→*A*, 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 window *λ*-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 *λ*⊢*nd*, 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 *S* _{ λ } 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 versions Open image in new window and Open image in new window defined as follows. For *A* with |*A*|=*nd*, we consider a vector space *V* _{ A } with a basis indexed by the elements of *A*. The choice of basis on *V* _{ A } 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 *I* _{ X },*S* _{ X } 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*=*Ver* _{ d } is the functor which associates with *V* the *d*-th Veronese embedding of \(\mathbb{P}V\).

## Example 3.6

(Covariants associated with graphs)

*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*=(

*nd*−

*e*,

*e*), as follows. Choose a labeling of the vertices of

*Q*with elements of [

*n*] and consider a \(\mbox{Young}^{d}_{n}\) tableau

*F*:

*D*

_{ λ }→[

*n*] having a column of size two with entries

*x*,

*y*(in some order) for each edge

*xy*∈

*E*(

*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

*Let* *Q*′ *be a subgraph of a graph* *Q*. *We have* Open image in new window *where* \(\mathcal{Q}\) *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 *σ* _{2}(*V*) be the variety of secant lines to *Ver* _{ d }(*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 *Ver* _{ d }(*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

*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≤

*j*≤

*μ*

_{1}

*μ*!=∏

_{ i }

*μ*

_{ i }!, we note that

*μ*!

*B*

_{1},…,

*B*

_{ m }of the Young tableau obtained by removing the first (

*v*−1) columns of

*S*, denote by

*l*

_{ i }(resp.

*h*

_{ i }) their lengths (resp. heights), let

*r*

_{ i }denote the length of the hook of

*μ*centered at (

*h*

_{ i },

*v*),

- (1)
\(\mu'_{v}>\mu'_{v+1}\): in which case \(m=\tilde{m}+1\),

*B*_{1}=*C*_{ v }(*S*), \(B_{i}=\tilde{B}_{i-1}\) for*i*>1. - (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 have*

*In particular*, (4.3)

*is equivalent to*

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

## Lemma 4.2

*If*1≤

*i*≠

*j*≤

*μ*

_{1}

*are such that*\(\mu'_{i}\leq\mu'_{j}\),

*then*

*As a consequence*,

*for*1≤

*j*<

*i*≤

*m*

*we have*

## Proof

*i*≠

*j*≤

*μ*

_{1}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

*B*

_{ i }. We have Using the fact that \(\mu'_{k}=h_{i}\) if \(k\in\mathcal{C}_{i}\), we get □

## Lemma 4.3

*For*1≤

*j*

_{1}<

*j*

_{2}<⋯<

*j*

_{ k }≤

*μ*

_{1},

*k*≥2,

*and*\(b_{i}\in C_{j_{i}}(S)\)

*a collection of entries lying in distinct columns of*

*S*,

*we let*

*σ*

*be the cyclic permutation*(

*a*,

*b*

_{ k },

*b*

_{ k−1},…,

*b*

_{1}).

*If*

*b*

_{1},

*b*

_{2}

*lie in different rows of*

*S*

*then*\(\mathfrak {c}_{\mu}(S)\cdot\sigma\cdot\mathfrak{c}_{\mu}(S)=0\).

*Otherwise*,

*letting*

*τ*=(

*a*,

*b*

_{ k },

*b*

_{ k−1},…,

*b*

_{2}),

*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*,

## Proof

*b*

_{1},

*b*

_{2}lie in distinct rows of

*S*, say

*b*

_{2}∈

*R*

_{ s }(

*S*), and let

*x*=

*S*(

*s*,

*j*

_{1}). Since

*x*≠

*b*

_{1},

*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

*x*=

*σ*(

*x*) and

*b*

_{1}=

*σ*(

*b*

_{2}).

*b*

_{1},

*b*

_{2}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*⋅(

*b*

_{1},

*b*

_{2}), we obtain

*r*

_{1}, to

*k*≤

*m*, 2≤

*j*

_{2}<⋯<

*j*

_{ k }≤

*m*, and \(b_{i}\in B_{j_{i}}\) for 2≤

*i*≤

*k*. Letting

*j*

_{1}=1, we have by the first part of the lemma that if

*b*

_{1}is not in the same row as

*b*

_{2}, and (with the previous notation for

*τ*and

*σ*)

*b*

_{1}and

*b*

_{2}are in the same row of

*S*. Since there are exactly

*r*

_{1}=

*l*

_{1}elements

*b*

_{1}∈

*B*

_{1}lying in the same row as

*b*

_{2}, the conclusion follows by summing (4.12) over all such

*b*

_{1}’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 *x* _{1}=*z* _{ v }, *r* _{1}=1, \(x_{i}=\tilde{x}_{i-1}\) and \(r_{i}=\tilde{r}_{i-1}\) for *i*>1. We may then assume that \(\mu'_{v}=\mu'_{v+1}\).

*i*>1, it is enough to show that

*v*+1≤

*j*≤

*v*+

*l*

_{1}−1, we get from (4.9) that

*v*+1≤

*j*≤

*v*+

*l*

_{1}−1. We have where the second to last equality follows from Lemma 4.3 and the fact that for each

*b*

_{2}∈

*C*

_{ j }(

*S*) there exists exactly one

*b*

_{1}∈

*C*

_{ v }(

*S*) situated in the same row. □

## Proof of Theorem 1.2

^{ m }

*r*

_{1}⋯

*r*

_{ m }, to

*Q*(

*X*) (=

*Q*

_{ m }) with the property

## Lemma 4.4

*If*

*z*

_{ j }

*is as in*(4.1)

*for some*

*j*<

*v*,

*and*\(\sigma\in \mathfrak{S} _{n}\)

*is a permutation that fixes all the entries of*

*S*

*contained in its first*(

*v*−1)

*columns*,

*then*

## Proof

*σ*(

*a*)≠

*a*, it follows that

*σ*(

*a*) is contained in

*C*

_{ i }(

*S*) for some

*i*with \(\mu'_{i}\leq\mu'_{j}\). Since

*σ*(

*a*)=

*a*, so

*σ*⋅(1−

*z*

_{ j })⋅

*σ*

^{−1}=(1−

*z*

_{ j }) 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

*τ*appearing in the expansion of \(\mathfrak{c}_{\mu}(S)\), i.e.

*τ*=

*τ*

_{1}

*τ*

_{2}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

*R*

_{ u }(

*T*)∩

*τ*(

*C*

_{ j }(

*S*)∪{

*a*})|≥2. Since

*τ*(

*a*)=

*a*and

*a*∈

*R*

_{ u }(

*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

*C*

_{ j }(

*S*), so

*b*∈

*τ*

_{2}(

*C*

_{ j }(

*S*)). Since \(\tau_{1}\in\mathcal{R}_{S}\) it follows that

*τ*

_{1}(

*b*)∈

*R*

_{ u }(

*S*)⊂

*R*

_{ u }(

*T*). It follows that the intersection

*R*

_{ u }(

*T*)∩

*τ*(

*C*

_{ j }(

*S*)∪{

*a*}) contains {

*a*,

*τ*

_{1}(

*b*)}, as desired. □

## Lemma 4.5

*Consider*\(\sigma\in\mathfrak{S}_{n}\)

*with*

*σ*(

*a*)=

*a*,

*and for*

*z*

_{ j }

*as in*(4.1)

*let*

*We have*

*σ*⋅

*Z*=

*Z*⋅

*σ*,

*and as a consequence*\(\mathfrak{c}_{\mu}(S)\cdot Z=Z\cdot\mathfrak{c}_{\mu}(S)\).

## Proof

*σ*⋅

*Z*=

*Z*⋅

*σ*follows from the fact that

*σ*in the expansion of \(\mathfrak{c}_{\mu}(S)\) satisfy

*σ*(

*a*)=

*a*. □

## Lemma 4.6

## Proof

*P*(

*X*)=

*X*

^{ t }. We argue by induction on

*t*: for

*t*=0, the result follows from (2.6). Assume now that

*t*>0. Expanding

*X*

^{ t−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

*α*

_{ μ }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. □

*t*=1,…,

*m*(recall the definition of

*l*

_{ i },

*h*

_{ i }and

*r*

_{ i }from (4.5))

*X*

_{ m }=

*X*(as defined in (4.16)), that

*Q*

_{ t }is a polynomial in

*X*

_{ t }, and that the definition of

*P*

_{ m }coincides with that in (4.14). For an element \(f\in\mathbb{K}[\mathfrak{S}_{n}]\), we define its right annihilator by

*I*is a right ideal,

*f*≡

*g*implies

*fh*≡

*gh*, but in general \(hf\not\equiv hg\). However, for any polynomial

*P*(

*X*) (where

*X*is as defined in (4.16)) we have

*α*

_{ μ }and using (4.20) to

We will show that *P* _{ m }≡*Q*(*X*) for some polynomial *Q* (namely *Q*(*X*)=*Q* _{ m }), 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

*With*≡

*as defined in*(4.23),

*the following relations hold for*1≤

*t*≤

*m*:

- (a
_{ t }) -
*P*_{ t }≡*Q*_{ t }. - (b
_{ t }) -
*P*_{ t }⋅(*X*_{ t }+*h*_{1})≡(*X*_{ t }+*h*_{1})⋅*P*_{ t }≡0.

Before that, we formulate some preliminary results.

## Lemma 4.8

*For*2≤

*j*≤

*m*

*and*

*t*≥1

*we have*

*If*

*Q*(

*X*

_{ t })

*is any polynomial in*

*X*

_{ t }

*and*

*t*+1≤

*j*≤

*m*,

*then*

*In particular*,

*for*

*t*+1≤

*j*≤

*m*

*we have*

## 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).

*j*>

*t*, we have by (4.10) that

*Q*

_{ t }is a polynomial in

*X*

_{ t }divisible by \((X_{t}-s_{t}^{t})\) (see (4.22)). □

## Lemma 4.9

*For*2≤

*j*≤

*m*

*and*

*t*≥1

*we have*

*If*

*Q*(

*X*

_{ t })

*is any polynomial in*

*X*

_{ t }

*and*

*t*+1≤

*j*≤

*m*,

*then*

*In particular*,

*for*

*t*+1≤

*j*≤

*m*

*we have*

## Proof

This follows from Lemma 4.8 using (4.25). □

## Lemma 4.10

*If*

*Q*

*is a polynomial then*

## Proof

*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

*t*, showing that (a

_{ t })⇒(b

_{ t }) and (a

_{ t }),(b

_{ t })⇒(a

_{ t+1}). When

*t*=1, we have \(r_{1}=s_{1}^{1}=l_{1}\) so

*P*

_{1}=

*Q*

_{1}=

*x*

_{1}−

*l*

_{1}and (a

_{1}) holds.

- Open image in new window :
- We have
- Open image in new window :
- We have(4.34)

*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

*Q*(

*X*), so (4.35) is equivalent to

*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

*i*≤

*t*−1, we obtain

## 5 Proof of Theorem 1.1

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

*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):

*m*

_{ 1 }=

*α*

_{ μ }. By Theorem 1.2

*n*

_{ 1 }=

*α*

_{ δ }, and, moreover,

*n*

_{ τ }≠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

*τ*∈

*L*(

*T*;

*U*) and

*σ*∈

*L*(

*U*;

*S*), then for every

*s*∈

*S*either (

*τ*⋅

*σ*)(

*s*)=

*s*or (

*τ*⋅

*σ*)(

*s*) lies strictly to the left of

*s*. Assume now that (

*τ*⋅

*σ*)(

*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

*U*∖

*S*, so

*τ*=

**1**.

*τ*≠

**1**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

*τ*≠

**1**, there exists an entry

*s*in

*S*such that

*τ*(

*s*)=

*a*is the unique entry of

*T*outside

*U*(

*s*=

*b*

_{1}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

*C*

_{ v }(

*T*) and in

*R*

_{ i }(

*τ*⋅

*S*), 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

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