Products of stabilizing representations

TL;DR

This paper investigates how products of stabilizing symmetric group representations follow specific recursive relations, introducing new difference operators to describe their behavior.

Contribution

It introduces a novel class of difference operators to characterize recursive relations in products of stabilizing representations.

Findings

Products satisfy recursive relations.

New difference operators effectively describe these relations.

Advances understanding of representation stability.

Abstract

We study representation stability in the sense of Church and Farb. We show that products of stabilizing Sn -representations fulfill certain recursive relations which can be described by a new class of difference operators.

Equations

$${\Delta }_m^{\lambda }:{\Lambda }_{\mathbb{N}}^{n_0,|\lambda |/m}\to {\Lambda }_{\mathbb{N}}^{n_0,|\lambda |/m},\{f_n{\}}_{n\ge 0}\mapsto \{{\Delta }_m^{\lambda }{f}_{n+m}{\}}_{n\ge 0}$$

$$\text{where Δmλfn=fn−(fn−m+λ) for all n≥m}.$$

$${\Delta }^{(2)}{\Delta }^{(1,1)}(s_{(n+{\alpha }_1,{\lambda }_1)}{s}_{(n+{\alpha }_2,{\lambda }_2)})=0\text{for all n>∣λ1∣+∣λ2∣+1−α2}.$$

$${\Delta }_2^{(3,3)}{\Delta }^{(3)}{\Delta }^{(2,1)}{\Delta }^{(1,1,1)}(s_{(n+{\alpha }_1,{\lambda }_1)}{s}_{(n+{\alpha }_2,{\lambda }_2)}{s}_{(n+{\alpha }_3,{\lambda }_3)})=0$$

$$\text{for all n>max{4,α1−α2+l(λ1)}+2(∣λ1∣+∣λ2∣+∣λ3∣+1)−α3}.$$

$${\Delta }_3^{(4,4,4)}{\Delta }_2^{(3,3,2)}{\Delta }^{(4)}{\Delta }^{(3,1)}({\Delta }^{(2,2)}{)}^2({\Delta }^{(2,1,1)}{)}^2{\Delta }^{(1,1,1,1)}(s_{(n+{\alpha }_1,{\lambda }_1)}{s}_{(n+{\alpha }_2,{\lambda }_2)}{s}_{(n+{\alpha }_3,{\lambda }_3)}{s}_{(n+{\alpha }_4,{\lambda }_4)})=0$$

$$\text{for sufficiently large n}.$$

$${\Delta }^{(1)}(f_n)=0\text{for all n>N}.$$

$${\Delta }^{(2)}{\Delta }^{(1,1)}(f_n^{(1)}{f}_n^{(2)})=0$$

$$\text{for all n>max{N1,N2,m1+m2+N1+N2+max{N1,N2}−2}}.$$

$${\Delta }_2^{(3,3)}{\Delta }^{(3)}{\Delta }^{(2,1)}{\Delta }^{(1,1,1)}(f_n^{(1)}{f}_n^{(2)}{f}_n^{(3)})=0$$

$$\text{for all n>max{N1,N2,2(m1+m2+m3+max{m1,m2,m3})+3(N1+N2+N3)−7}}.$$

$$\Delta_{3}^{(4,4,4)}\Delta_{2}^{(3,3,2)}\Delta^{(4)}\Delta^{(3,1)}(\Delta^{(2,2)})^{2}(\Delta^{(2,1,1)})^{2}\Delta^{(1,1,1,1)}\big{(}f^{(1)}_{n}f^{(2)}_{n}f^{(3)}_{n}f^{(4)}_{n}\big{)}=0$$

$$\text{for sufficiently large n}.$$

$$({\Delta }_{|{\lambda }_1|/k}^{{\lambda }_1}{)}^{q_1}\cdots ({\Delta }_{|{\lambda }_r|/k}^{{\lambda }_r}{)}^{q_r}\left(f_n^{(1)}\cdots f_n^{(k)}\right)=0\text{for all n>N}$$

$$({\Delta }_{|{\lambda }_1|/k}^{{\lambda }_1}{)}^{q_1}\cdot ({\Delta }_{|{\lambda }_r|/k}^{{\lambda }_r}{)}^{q_r}\left(s_{(n+{\alpha }_1,{\mu }_1)}\cdots s_{(n+{\alpha }_k,{\mu }_k)}\right)=0\text{for all n>N}$$

$${\Delta }_m^{\lambda }{\Delta }_l^{\mu }(f_n)={\Delta }_l^{\mu }{\Delta }_m^{\lambda }(f_n)=\text{for all n}.$$

$${\Delta }_m^{\lambda }{\Delta }_l^{\mu }{f}_n={\Delta }_m^{\lambda }(f_n-(f_{n-l}+\mu ))=f_n-(f_{n-l}+\mu )-(f_{n-m}+\lambda )+(f_{n-l-m}+\mu +\lambda )$$

$$={\Delta }_l^{\mu }(f_n-(f_{n-m}+\lambda ))={\Delta }_l^{\mu }{\Delta }_m^{\lambda }{f}_n.$$

$${\Delta }^{(2)}{\Delta }^{(1,1)}(s_{(n+{\alpha }_1,{\lambda }_1)}{s}_{(n+{\alpha }_2,{\lambda }_2)})=0\text{for all n>∣λ1∣+∣λ2∣+1−min{α1,α2}}.$$

$$|{\lambda }_1|+|{\lambda }_2|+1-\min \{{\alpha }_1,{\alpha }_2\}\le m_1+m_2+N_1+N_2+\max \{N_1,N_2\}-2.$$

$${\Delta }_2^{(3,3)}{\Delta }^{(3)}{\Delta }^{(2,1)}{\Delta }^{(1,1,1)}(s_{(n+{\alpha }_1,{\lambda }_1)}{s}_{(n+{\alpha }_2,{\lambda }_2)}{s}_{(n+{\alpha }_3,{\lambda }_3)})=0$$

$$\text{for all n>max{4,α1−α2+l(λ1)}+2(∣λ1∣+∣λ2∣+∣λ3∣+1)−α3}.$$

$$\max \{4,{\alpha }_1-{\alpha }_2+l({\lambda }_1)\}+2(|{\lambda }_1|+|{\lambda }_2|+|{\lambda }_3|+1)-{\alpha }_3$$

$$\le m_1+N_2-1+m_1+N_1-1+2(m_1+m_2+m_3+N_1+N_2+N_3-2)+N_3-1$$

$$=4m_1+2m_2+2m_3+3(N_1+N_2+N_3)-7$$

$${\Delta }_{|{\lambda }_1|/k}^{{\lambda }_1}...{\Delta }_{|{\lambda }_m|/k}^{{\lambda }_m}(s_{(n+{\alpha }_1)}\cdots s_{(n+{\alpha }_{k-1})}{s}_{(n)}{s}_{\beta })=0$$

$${\Delta }_{|{\lambda }_1|/k}^{{\lambda }_1}...{\Delta }_{|{\lambda }_1|/k}^{{\lambda }_m}(s_{(n+{\alpha }_1,{\mu }_1)}\cdots s_{(n+{\alpha }_{k-1},{\mu }_{k-1})}{s}_{(n,{\mu }_k)})=0$$

$$\prod _{q=1}^k{s}_{(n+{\alpha }_q,{\mu }_q)}$$

$$=\prod _{q=1}^k\det \left(\begin{array}{cccc}{s}_{(n+{\alpha }_q)}& s_{(n+{\alpha }_q+1)}& ...& s_{(n+{\alpha }_q+l({\mu }_q))}\\ s_{({\mu }_{q,1}-1)}& s_{({\mu }_{q,1})}& ...& s_{({\mu }_{q,1}+l({\mu }_q)-1)}\\ ...& ...& ...& ...\\ s_{({\mu }_{q,l({\mu }_q)}-l({\mu }_q))}& ...& ...& s_{({\mu }_{q,l({\mu }_q)})}\end{array}\right)$$

Full text

1. Introduction

We study representations of the symmetric group $S_{n}$ and formulate our statements about $S_{n}$-representations in the world of symmetric functions. We refer to Macdonald [2] for background on $S_{n}$-representations and symmetric functions. The ${\mathbb{Z}}$-module of symmetric functions corresponding to (virtual) $S_{n}$-representations is denoted by $\Lambda_{n}$. A number partition $\lambda$ of $n\in{\mathbb{N}}$ is a finite weakly decreasing sequence $\lambda=(\lambda_{1},...,\lambda_{l})$ of positive integers such that $\lambda_{1}+...+\lambda_{l}=n$. We also write $\lambda\vdash n$ or $|\lambda|=n$ in that case. The number $l=l(\lambda)$ is the length of $\lambda$. The set of Schur functions $\{s_{\lambda}~{}|~{}\lambda\vdash n\}$ is a basis of $\Lambda_{n}$. We define the componentwise sum of two partitions $\lambda=(\lambda_{1},...,\lambda_{l})$ and $\mu=(\mu_{1},...,\mu_{k})$ with $l\leq k$ by $\lambda+\mu=(\lambda_{1}+\mu_{1},...,\lambda_{l}+\mu_{l},\mu_{l+1},...,\mu_{k})$. For a fixed partition $\lambda$ we denote by $s_{\mu}+\lambda$ the function $s_{\mu+\lambda}$ and extend this definition from the basis of Schur functions linearly to all symmetric functions. By $\Lambda_{{\mathbb{N}}}^{n_{0},k}$ we denote the ${\mathbb{Z}}$-module of sequences $\{f_{n}\}_{n\in{\mathbb{N}}}$ with $f_{n}\in\Lambda_{nk+n_{0}}$ for all $n\in{\mathbb{N}}$. For every $n_{0}\in{\mathbb{N}}$, every partition $\lambda$ and every divisor $m$ of $|\lambda|$ we define

[TABLE]

[TABLE]

We write $\Delta^{\lambda}$ for $\Delta^{\lambda}_{1}$. For a partition $\lambda=(\lambda_{1},...,\lambda_{l})$ and $n\geq\lambda_{1}$ we write$(n,\lambda)$ for $(n,\lambda_{1},\lambda_{2},...,\lambda_{l})$. We consider sequences of Schur functions of the form $\{s_{(n,\lambda)}\}_{n\geq\lambda_{1}}\in\Lambda_{{\mathbb{N}}}^{|\lambda|,1}$. Let $\alpha_{1},...,\alpha_{k}\in{\mathbb{N}}$ and $\lambda_{1},...,\lambda_{k}$ be number partitions. Then the sequence of products $\{s_{(n+\alpha_{1},\lambda_{1})}\cdots s_{(n+\alpha_{k},\lambda_{k})}\}_{n}$ is an element of $\Lambda_{{\mathbb{N}}}^{\alpha_{1}+|\lambda_{1}|+\cdots\alpha_{k}+|\lambda_{k}|,k}$. We can apply difference operators $\Delta_{|\mu|/k}^{\mu}$ on it where $|\mu|$ is a multiple of $k$. Now, we are in position to formulate our main theorem.

Theorem 1.0**.**

Let $\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}\geq\geq 0$ and $\lambda_{1},\lambda_{2},\lambda_{3}$ be number partitions. Then

• (a)

[TABLE]

The set $\{\Delta^{(2)},\Delta^{(1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

• (b)

[TABLE]

[TABLE]

The set $\{\Delta_{2}^{(3,3)},\Delta^{(3)},\Delta^{(2,1)},\Delta^{(1,1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

There is experimental evidence that an analogous statement about fourfold products holds. We formulate this in the following conjecture. We do not provide a complete proof of this statement but show how our methods indicate its validity until reaching a point where the number of cases to discuss is massive.

Conjecture 1.0**.**

Let $\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}\geq\alpha_{4}\geq 0$ and $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ be number partitions. Then

[TABLE]

[TABLE]

The multiset $\{\Delta_{3}^{(4,4,4)},\Delta_{2}^{(3,3,2)},\Delta^{(4)},\Delta^{(3,1)},\Delta^{(2,2)},\Delta^{(2,2)},\Delta^{(2,1,1)},\Delta^{(2,1,1)},\Delta^{(1,1,1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

We show in Lemma 2.1 that every difference operator $\Delta_{m}^{\lambda}$ is linear so that we can expand the theorem to a wider set of symmetric function sequences. A sequence $\{f_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{n_{0},1}$ stabilizes at $N\in{\mathbb{N}}$ in the sense of Church and Farb [1] if

[TABLE]

We get the following corollary.

Corollary 1.0**.**

Let $m_{1},m_{2},m_{3},N_{1},N_{2},N_{3}\in{\mathbb{N}}$ and $\{f^{(1)}_{n}\}_{n},\{f^{(2)}_{n}\}_{n},\{f^{(3)}_{n}\}_{n}$ be sequences such that $\{f^{(i)}_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{m_{i},1}$ stabilizes at $N_{i}$ for all $i\in\{1,2,3\}$. Then

• (a)

[TABLE]

[TABLE]

The set $\{\Delta^{(2)},\Delta^{(1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

• (b)

[TABLE]

[TABLE]

The set $\{\Delta_{2}^{(3,3)},\Delta^{(3)},\Delta^{(2,1)},\Delta^{(1,1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

In the same way, Conjecture 1.0 is equivalent to the following statement.

Conjecture 1.0**.**

Let $m_{1},...,m_{4},N_{1},...,N_{4}\in{\mathbb{N}}$ and $\{f^{(1)}_{n}\}_{n},\{f^{(2)}_{n}\}_{n},\{f^{(3)}_{n}\}_{n},\{f^{(4)}_{n}\}_{n}$ be sequences such that $\{f^{(i)}_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{m_{i},1}$ stabilizes at $N_{i}$ for all $i\in\{1,2,3,4\}$. Then

[TABLE]

[TABLE]

The multiset $\{\Delta_{3}^{(4,4,4)},\Delta_{2}^{(3,3,2)},\Delta^{(4)},\Delta^{(3,1)},\Delta^{(2,2)},\Delta^{(2,2)},\Delta^{(2,1,1)},\Delta^{(2,1,1)},\Delta^{(1,1,1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

We formulate a statement about products of arbitrariely many stabilizing sequences as a question.

Question 1.0**.**

Let $k\geq 1$ and $\{f^{(1)}_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{m_{1},1},...,\{f^{(k)}_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{m_{k},1}$ be stabilizing sequences. Let $\{\lambda_{1},...,\lambda_{r}\}$ be the set of partitions of the numbers $k,2k,...,(k-1)k$. Is there a set of nonnegative integers $\{q_{1},...,q_{k-1}\}$ and a number $N$ such that

[TABLE]

and $q_{1}+...+q_{k-1}$ is minimal with this property and how can we compute the numbers $q_{1},...,q_{r}$ and $N$?

By Lemma 2.1 (ii) this is equivalent to

Question 1.0**.**

Let $k\geq 1$, $\alpha_{1}\geq\dots\geq\alpha_{4}\geq 0$ and $\mu_{1},\dots,\mu_{k}$ be number partitions. We consider the sequence $\{s_{(n+\alpha_{1},\mu_{1})}\cdots s_{(n+\alpha_{k},\mu_{k})}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+|\mu_{1}|+\cdots+\alpha_{k}+|\mu_{k}|,k}$. Let $\{\lambda_{1},...,\lambda_{r}\}$ be the set of partitions of the numbers $k,2k,...,(k-1)k$. Is there a set of nonnegative integers $\{q_{1},...,q_{k-1}\}$ and a number $N$ such that

[TABLE]

and $q_{1}+...+q_{k-1}$ is minimal with this property and how can we compute the numbers $q_{1},...,q_{r}$ and $N$?

The rest of the paper is dedicated to the proof of Theorem 1.0 and Corollary 1.0.

2. Reduction to homogeneous symmetric functions

We show in the following lemma that the difference operators commute such that we are free to choose their order and that they are linear.

Lemma 2.1**.**

Let $\lambda$ and $\mu$ be partitions, $m$ a divisor of $|\lambda|$ and $l$ a divisor of $|\mu|$ with $|\mu|/l=|\lambda|/m$. Let $n_{0}\geq 0$ and $\{f_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{n_{0},|\lambda|/m}$. Then

• (i)

[TABLE]

• (ii)

The map $\Delta_{m}^{\lambda}:\Lambda_{{\mathbb{N}}}^{n_{0},|\lambda|/m}\rightarrow\Lambda_{{\mathbb{N}}}^{n_{0},|\lambda|/m}$ is linear.

Proof.

(i) We have

[TABLE]

[TABLE]

(ii) $\Delta_{m}^{\lambda}$ is the difference of the shift operator $\{f_{n}\}_{n}\mapsto\{f_{n+m}\}_{n}$ which is linear and the map $\{f_{n}\}_{n}\mapsto\{f_{n}+\lambda\}_{n}$ which is a linear extension. $\blacksquare$

It follows from Lemma 2.1 (i) that we can define iterated products $\prod_{\Delta\in D}\Delta$ over sets of difference operators $D=\{\Delta_{m_{1}}^{\lambda_{1}},\dots,\Delta_{m_{r}}^{\lambda_{r}}\}$. Now, we prove Corollary 1.0 using Theorem 1.0 and Lemma 2.1 (ii).

Proof.

(a) The sequence $\{f_{n}^{(i)}\}_{n}\in\Delta_{{\mathbb{N}}}^{m_{i},1}$ stabilizes at $N_{i}$ for every $i\in\{1,2\}$. It follows that $f_{n}^{(i)}$ is a linear combination of Schur functions $s_{(n+\alpha_{i},\lambda_{i})}$ with $\alpha_{i}\geq 1-N_{i}$ and $\lambda_{i}\vdash m_{i}-\alpha_{i}$ for all $n\geq N_{i}$. Theorem 1.0 yields that

[TABLE]

We have

[TABLE]

The claim follows from Lemma 2.1 (ii).

(b) The sequence $\{f_{n}^{(i)}\}_{n}\in\Delta_{{\mathbb{N}}}^{m_{i},1}$ stabilizes at $N_{i}$ for every $i\in\{1,2,3\}$. It follows that $f_{n}^{(i)}$ is a linear combination of Schur functions $s_{(n+\alpha_{i},\lambda_{i})}$ with $m_{i}\geq\alpha_{i}\geq 1-N_{i}$ and $\lambda_{i}\vdash m_{i}-\alpha_{i}$ for all $n\geq N_{i}$. Suppose $\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}$. Theorem 1.0 yields that

[TABLE]

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

The claim follows from Lemma 2.1 (ii). $\blacksquare$

We want to show next that we can restrict to products of homogeneous symmetric functions $s_{(n)}$ if we additionally multiply with a constant sequence.

Lemma 2.2**.**

Let $k\in{\mathbb{N}}$ and $\lambda_{1},...,\lambda_{m}$ be number partitions of multiples of $k$. If for all number partitions $\beta$ and $\alpha_{1},...,\alpha_{k-1}\in{\mathbb{N}}$ the sequence $\{s_{(n+\alpha_{1})}\cdots s_{(n+\alpha_{k-1})}s_{(n)}s_{\beta}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+\cdots+\alpha_{k-1}+|\beta|,k}$ fulfills

[TABLE]

*for all $n$ greater than some number $n_{0}(\alpha_{1},...,\alpha_{k-1},|\beta|)$ then for all number partitions $\mu_{1},...,\mu_{k}$ and $\alpha_{1},...,\alpha_{k-1}\in{\mathbb{N}}$ the sequence

$\{s_{(n+\alpha_{1},\mu_{1})}\cdots s_{(n+\alpha_{k-1},\mu_{k-1})}s_{(n,\mu_{k})}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+\cdots+\alpha_{k-1}+|\mu_{1}|+\cdot+|\mu_{k}|,k}$ fulfills*

[TABLE]

for all n>\max\{n_{0}(\alpha_{1}+i_{1},...,\alpha_{k}+i_{k},|\mu_{1}|-i_{1}+...+|\mu_{k}|-i_{k})~{}|~{}i_{q}\in\{0,...,l(\mu_{q})\}~{}\text{for all q\in{1,...,k}}\}.

Proof.

Let $\alpha_{k}=0$. The Jacobi-Trudi identity yields

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where for $\gamma\in\{\mu_{1},...,\mu_{k}\}$ the matrix $M_{\gamma,i}$ is the matrix we get by deleting the $i$th column of

[TABLE]

The degree of $\prod_{q=1}^{k}\det(M_{\mu_{q},i_{q}})$ is $|\mu_{1}|-i_{1}+...+|\mu_{k}|-i_{k}$. It follows from the assumption that this sequence vanishes under $\Delta^{\lambda_{1}}_{n_{1}}...\Delta^{\lambda_{m}}_{n_{m}}$ for n>\max\{n_{0}(\alpha_{1}+i_{1},...,\alpha_{k}+i_{k},|\mu_{1}|-i_{1}+...+|\mu_{k}|-i_{k})~{}|~{}i_{q}\in\{0,...,l(\mu_{q})\}~{}\text{for all q\in{1,...,k}}\}. $\blacksquare$

It follows that to prove Theorem 1.0 it is sufficient to prove the following

Proposition 2.0**.**

Let $\alpha_{1},\alpha_{2},\alpha_{3}\in{\mathbb{N}}$ and $\beta$ be a number partition.

• (a)

The sequence $\{s_{(n+\alpha_{1})}s_{(n)}s_{\beta}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+|\beta|,2}$ fulfills

[TABLE]

• (b)

The sequence $\{s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)}s_{\beta}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+\alpha_{2}+|\beta|,3}$ fulfills

[TABLE]

[TABLE]

• (c)

The sequence $\{s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n+\alpha_{3})}s_{(n)}s_{\beta}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+\alpha_{2}+\alpha_{3}+|\beta|,4}$ fulfills

[TABLE]

[TABLE]

For every symmetric function $f$ and $l\in{\mathbb{N}}$ we write $f_{\leq l}$ for the part of the Schur function decomposition of $f$ with partitions of length less than or equal to $l$ and $f_{>l}$ for the part of the Schur function decomposition with partitions of length greater than $l$.

Lemma 2.3**.**

Let $k,n_{0},l\in{\mathbb{N}}$ and $\{f_{n}\}_{n}\in\Lambda_{{\mathbb{N}}}^{n_{0},k}$. Let $\lambda_{1},...,\lambda_{m}$ be number partitions of multiples of $k$. Suppose

[TABLE]

Then

[TABLE]

Proof.

We have

[TABLE]

[TABLE]

We can write

[TABLE]

for a function $g_{n}$ with only partitions of length greater than $l$ in its Schur function decomposition. It follows

[TABLE]

All partitions with length less than or equal to $l$ appear in $\left(\prod_{i:l(\lambda_{i})\leq l}\Delta^{\lambda_{i}}_{|\lambda_{i}|/k}\right)(f_{n})_{\leq l}$ while all partitions with length greater than $l$ appear in $\left(\prod_{i:l(\lambda_{i})\leq l}\Delta^{\lambda_{i}}_{|\lambda_{i}|/k}\right)g_{n}+\Delta^{\lambda_{1}}_{|\lambda_{1}|/k}...\Delta^{\lambda_{m}}_{|\lambda_{m}|/k}(f_{n})_{>l}$ and it follows that each of these two parts must itself be zero. $\blacksquare$

Consider a semistandard skew tableau $T$ of shape $\nu/\beta$ and weight $(n+\alpha_{1},n+\alpha_{2},...,n+\alpha_{k})$. We split $T$ into two parts: The part of the first $\beta_{1}$ columns which we call the centre of $T$ and denote it ${\mathrm{centre}}(T)$ and the rest which we call the arm of $T$ and denote it ${\mathrm{arm}}(T)$. For example, let

[TABLE]

Then

[TABLE]

For fixed $\beta$ and $k$, there are only finitely many tableaux appearing as centres of tableaux of shape $\nu/\beta$ for arbitrary $\nu$ and we denote this finite set by $c_{k}(\beta)$. We denote the set of semistandard Young tableaux of weight $\gamma$ and arbitrary shape by $ST(\gamma)$ and the number of occurences of the Symbol $i$ in the tableau $c\in c_{k}(\beta)$ by $c(i)$

Lemma 2.4**.**

Let $\alpha_{1},...,\alpha_{k}\in{\mathbb{N}}$ and $\beta$ be a number partition. Then

[TABLE]

for all $n\in{\mathbb{N}}$.

Proof.

We have

[TABLE]

where the sum runs over all semistandard skew tableau $T^{\prime}$ of shape $\nu/\beta$ for any $\nu$ and weight $(n+\alpha_{1},n+\alpha_{2},...,n+\alpha_{k})$ We can rewrite this sum by splitting every such skew tableau into its centre and arm:

[TABLE]

where the sum runs over all $T\in ST((n+\alpha_{1}-c(1),...,n+\alpha_{k}-c(k)))$ such that $T$ has as most as many rows as $c$ has rows of length $\beta_{1}$. Note that $\sum_{T}s_{{\mathrm{shape}}(T)}$ is the part of $s_{(n+\alpha_{1}-c(1))}...s_{(n+\alpha_{k}-c(k))}$ with Schur functions with partitions with at most as many rows as $c$ has rows of length $\beta_{1}$. $\blacksquare$

The following lemma follows from the previous two lemmas.

Lemma 2.5**.**

Let $k\in{\mathbb{N}}$ and $\beta$ be a number partition. Let $\lambda_{1},...,\lambda_{m}$ be number partitions of multiples of $k$. If for all numbers $\alpha_{1},...,\alpha_{k}\in{\mathbb{N}}$ there is a number $N(\alpha_{1},...,\alpha_{k})$ such that the sequence $\{s_{(n+\alpha_{1})}...s_{(n+\alpha_{k})}\}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+...+\alpha_{k},k}$ fulfills

[TABLE]

then the sequence $\{s_{(n+\alpha_{1})}...s_{(n+\alpha_{k})}s_{\beta}\}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+...+\alpha_{k}+|\beta|,k}$ fulfills

[TABLE]

[TABLE]

This lemma shows that it is sufficient to prove the following proposition.

Proposition 2.0**.**

Let $\alpha_{1}\geq\alpha_{2}\geq 0$.

• (a)

The sequence $\{s_{(n+\alpha_{1})}s_{(n)}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1},2}$ fulfills

[TABLE]

The set $\{\Delta^{(2)},\Delta^{(1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

• (b)

The sequence $\{s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+\alpha_{2},3}$ fulfills

[TABLE]

[TABLE]

The set $\{\Delta_{2}^{(3,3)},\Delta^{(3)},\Delta^{(2,1)},\Delta^{(1,1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

We do not provide a complete proof of the following statement about fourfold products but show how our methods point to its validity until reaching a point where the number of cases to discuss is massive.

Conjecture 2.0**.**

Let $\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}\geq 0$. The sequence $\{s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n+\alpha_{3})}s_{(n)}\}_{n}\in\Lambda_{{\mathbb{N}}}^{\alpha_{1}+\alpha_{2}+\alpha_{3},4}$ fulfills

[TABLE]

[TABLE]

The multiset $\{\Delta_{3}^{(4,4,4)},\Delta_{2}^{(3,3,2)},\Delta^{(4)},\Delta^{(3,1)},\Delta^{(2,2)},\Delta^{(2,2)},\Delta^{(2,1,1)},\Delta^{(2,1,1)},\Delta^{(1,1,1,1)}\}$ is minimal in the sense that the above sequence is not eventually zero if we remove one of the difference operators.

3. Proof of Proposition 2.0

In the whole section $n$ and $\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}$ are natural numbers. In the next lemmas, we use partial matrices of the form

[TABLE]

For every $i\in\{1,...,k-1\}$, we write $a_{i}$ for the $i$th row of $a$ and $|a_{i}|$ for the sum of the entries of the $i$th row. Let $k,n\in{\mathbb{N}}$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. Let $P_{k,n,\alpha}$ be the set of all partial matrices $\begin{pmatrix}a_{11}&a_{12}&a_{13}&\dots&a_{1k}\\ a_{21}&a_{22}&\dots&a_{2,(k-1)}\\ \vdots\\ a_{(k-1),1}&a_{(k-1),2}\\ a_{k,1}\end{pmatrix}$ with real entries and

[TABLE]

[TABLE]

[TABLE]

or written as vector inequalities and equalities:

[TABLE]

[TABLE]

$P_{k,n,\alpha}$ is a convex polytope in ${\mathbb{R}}^{\binom{k+1}{2}}$. If $M$ is a set of partial matrices we write $M^{{\mathbb{Z}}}$ for its subset of partial matrices with only integer valued entries.

Proposition 3.0**.**

Let $k,n\in{\mathbb{N}}$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. The polytope $P_{k,n,\alpha}$ is $\binom{k}{2}$-dimensional.

Proof.

$P_{k,n,\alpha}$ is contained in the affine subspace defined by

[TABLE]

The above matrix has rank $k$. This implies that the affine subspace has dimension $\binom{k+1}{2}-k=\binom{k}{2}$. Otherwise, $P_{k,n,\alpha}$ contains the following $\binom{k}{2}+1$ affine independent points. For every $i\in\{0,...,k-1\}$ we construct ${\mathrm{min}}\{1,i\}$ points starting with

[TABLE]

We get the next point by moving $n+\alpha_{k}$ diagonally left and down:

[TABLE]

Now, we move $n+\alpha_{k-1}$ and $n+\alpha_{k}$ diagonally left and down:

[TABLE]

We go on moving the rightmost nonzero values diagonally left and down until getting the point

[TABLE]

We constructed $1+1+2+\cdots(k-1)=\binom{k}{2}+1$ affine independent points lying in $P_{k,n,\alpha}$. $\blacksquare$

Lemma 3.1**.**

Let $k,n\in{\mathbb{N}}$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. Then

[TABLE]

Proof.

The product $\prod_{i=1}^{k}s_{(n+\alpha_{i})}$ is the homogeneous symmetric function $h_{(n+\alpha_{1},...,n+\alpha_{k})}$. It follows from the transition matrix between the basis of Schur functions and the basis of homogeneous symmetric functions that $h_{(n+\alpha_{1},...,n+\alpha_{k})}$ is the sum $\sum_{T}s_{{\mathrm{shape}}(T)}$ running over all semistandard Young tableaux $T$ of weight $(n+\alpha_{1},...,n+\alpha_{k})$. For every such tableau $T$ let $a_{i,j}(T)$ be the number of $(i+j-1)$’s in the $i$th row. Then the map given by

[TABLE]

is a bijection between the semistandard Young tableaux $T$ of weight $(n+\alpha_{1},...,n+\alpha_{k})$ and $P_{k,n,\alpha}^{{\mathbb{Z}}}$. $\blacksquare$

Lemma 3.2**.**

Let $k\geq 1$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. We consider the sequence $\{\prod_{i=1}^{k}s_{(n+\alpha_{i})}\}_{n}\in\Lambda_{{\mathbb{N}}}^{|\alpha|,k}$. For all $n\geq 1$, we have

[TABLE]

[TABLE]

Proof.

It follows from the previous lemma that

[TABLE]

There is an injection $P_{k,n-1,\alpha}^{{\mathbb{Z}}}\rightarrow P_{k,n,\alpha}^{{\mathbb{Z}}}$ given by

[TABLE]

The matrices that are not hit by this map are those with a [math] in the first column. This property is equivalent to $a_{k,1}=0$ because of $0\leq a_{k,1}\leq a_{k-1,1}\leq...\leq a_{11}$. $\blacksquare$

We look at the polytope $\Delta^{(1^{k})}P_{k,n,\alpha}:=\{A\in P_{k,n,\alpha}:~{}a_{k,1}=0\}$.

Proposition 3.0**.**

Let $k\geq 1$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. $\Delta^{(1^{k})}P_{k,n,\alpha}$ is a facet of $P_{k,n,\alpha}$.

Proof.

$\Delta^{(1^{k})}P_{k,n,\alpha}$ is a proper face of $P_{k,n,\alpha}$ because of $a_{k,1}=0$ for all $A\in\Delta^{(1^{k})}P_{k,n,\alpha}$ but not for every $A\in P_{k,n,\alpha}$ and $a_{k,1}\geq 0$ for all $A\in P_{k,n,\alpha}$. This face is $\left(\binom{k}{2}-1\right)$-dimensional because the set of $\binom{k}{2}+1$ many affine independent points of $P_{k,n,\alpha}$ given in the proof of Proposition 3.0 contains exactly one point that does not lie in $\Delta^{(1^{k})}P_{k,n,\alpha}$. $\blacksquare$

Lemma 3.3**.**

Let $k\geq 1$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. We consider the sequence $\{\prod_{i=1}^{k}s_{(n+\alpha_{i})}\}_{n}\in\Lambda_{{\mathbb{N}}}^{|\alpha|,k}$. For all $n\geq 2$, we have

[TABLE]

[TABLE]

Proof.

Lemma 3.2 yields

[TABLE]

[TABLE]

There is an injective map $\{A\in P_{k,n-1,\alpha}^{{\mathbb{Z}}}:~{}a_{k,1}=0\}\mapsto\{A\in P_{k,n,\alpha}^{{\mathbb{Z}}}:~{}a_{k,1}=0\}$ given by

[TABLE]

The matrices that are not hit are those with $a_{i1}=0$ for a $1\leq i\leq k-1$ or $a_{1k}=0$. We can reduce the condition to $a_{1k}=0$ or $a_{k-1,1}=0$ because of $0\leq a_{k-1,1}\leq a_{k-2,1}\leq...\leq a_{11}$. $\blacksquare$

Let $\Delta^{(2,1^{k-2})}\Delta^{(1^{k})}P_{k,n,\alpha}=\{A\in\Delta^{(1^{k})}P_{k,n,\alpha}:~{}a_{1k}=0\vee a_{k-1,1}=0\}$.

Proposition 3.0**.**

Let $k\geq 1$ and $\alpha=(\alpha_{1},...,\alpha_{k})\in{\mathbb{N}}_{0}^{k}$ with $\alpha_{1}\geq...\geq\alpha_{k}$. $\Delta^{(2,1^{k-2})}\Delta^{(1^{k})}P_{k,n,\alpha}$ is the union of the two facets $\{A\in\Delta^{(1^{k})}P_{k,n,\alpha}:~{}a_{1,k}=0\}$ and $\{A\in\Delta^{(1^{k})}P_{k,n,\alpha}:~{}a_{k-1,1}=0\}$ of $\Delta^{(1^{k})}P_{k,n,\alpha}$.

Proof.

The two sets are proper subsets of $\Delta^{(1^{k})}P_{k,n,\alpha}$ and faces because of $a_{1,k}\geq 0$ and $a_{k-1,1}\geq 0$ for all $A\in\Delta^{(1^{k})}P_{k,n,\alpha}$. $\{A\in\Delta^{(1^{k})}P_{k,n,\alpha}:~{}a_{k-1,1}=0\}$ is a facet of $\Delta^{(1^{k})}P_{k,n,\alpha}$ because the set of $\binom{k}{2}+1$ many affine independent points of $P_{k,n,\alpha}$ given in the proof of Proposition 3.0 contains exactly two points with $a_{k-1,1}>0$. For $\{A\in\Delta^{(1^{k})}P_{k,n,\alpha}:~{}a_{1,k}=0\}$ we slightly modify the list of affine independent points given in the proof of Proposition 3.0. For every $i\in\{2,...,k-2\}$ we take the $i$ points

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We additionally take the $k-1$ points

[TABLE]

and

[TABLE]

These are $2+3+...+(k-2)+(k-1)=\binom{k}{2}-1$ many affine independent points. $\blacksquare$

Now we can prove Proposition 2.0.

Proof of Proposition 2.0.

$(a)$ It follows from Lemma 3.3 that for all $n\geq 2$:

[TABLE]

[TABLE]

running over all

[TABLE]

with

[TABLE]

[TABLE]

But this cannot be for $n>0$. Therefore the sum is zero.

We show the minimality of the set $\{\Delta^{(2)},\Delta^{(1,1)}\}$ next. By Lemma 3.2, we have

[TABLE]

[TABLE]

running over all

[TABLE]

with

[TABLE]

This is the matrix

[TABLE]

and the above sum is not zero. The term $\Delta^{(2)}s_{(n+\alpha_{1})}s_{(n)}$ does also not equal zero because the sum

[TABLE]

has the summand $s_{(n+\alpha_{1},n)}$ that is not cancelled by applying $\Delta^{(2)}$ because $|a_{1}|\geq n-1+\alpha_{1}$ for all $A\in P_{2,n-1,\alpha}^{{\mathbb{Z}}}$.

$(b)$ It follows from Lemma 3.3 that for all $n\geq 2$:

[TABLE]

[TABLE]

running over all

[TABLE]

with

[TABLE]

[TABLE]

We want to apply the map $\Delta^{(3)}$ next. We treat the two sets $Q_{n,21}=\{A\in P_{3,n-1,\alpha}^{{\mathbb{Z}}}~{}|~{}a_{31}=0\wedge a_{21}=0\}$ and $Q_{n,13}=\{A\in P_{3,n-1,\alpha}^{{\mathbb{Z}}}~{}|~{}a_{31}=0\wedge a_{13}=0\wedge a_{21}>0\}$ separately. There is an injection $Q_{n-1,21}\rightarrow Q_{n,21}$ given by

[TABLE]

The only matrix that is not hit by this map is $\begin{pmatrix}n+\alpha_{1}&n+\alpha_{2}&0\\ 0&n\\ 0\end{pmatrix}$. It follows

[TABLE]

[TABLE]

We have

[TABLE]

It follows that for every $B\in Q_{n-1,13}$ there is exactly one $A\in Q_{n,13}$ with $(|a_{1}|,|a_{2}|)=(|b_{1}|+3,|b_{2}|)$. It is the matrix $A$ with $a_{21}=b_{21}-1$. There is one matrix in $Q_{n-1,13}$ and one or two matrices in $Q_{n,13}$ not involved in this correspondence depending on the parity of $n+|\alpha|$. These matrices are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now in $\Delta^{(3)}\Delta^{(2,1)}\Delta^{(1,1,1)}(s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)})$, the Schur function $s_{(2n+|\alpha|,n)}$ from before is subtracted and what is left is

[TABLE]

[TABLE]

Applying $\Delta_{2}^{(3,3)}$ to this yields [math].

We show the minimality of the set $\{\Delta^{(3,3)}_{2},\Delta^{(3)},\Delta^{(2,1)},\Delta^{(1^{3})}\}$ next. We see above that $\Delta^{(3)}\Delta^{(2,1)}\Delta^{(1,1,1)}(s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)})$ is not zero. The term $\Delta^{(3,3)}_{2}\Delta^{(3)}\Delta^{(2,1)}(s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)})$ is not zero because the summand $s_{(n+\alpha_{1},n+\alpha_{2},n)}$ in the Schur function expansion of $s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)}$ is not cancelled by $\Delta^{(3)}$ or $\Delta^{(2,1)}$ because $|a_{1}|\geq n-1+\alpha_{2}$ for all $A\in P_{3,n-1,\alpha}^{{\mathbb{Z}}}$ and it is not cancelled by $\Delta^{(3,3)}_{2}$ because $|a_{1}|\geq n-2+\alpha_{2}$ for all $A\in P_{3,n-2,\alpha}^{{\mathbb{Z}}}$. The term $\Delta^{(3,3)}_{2}\Delta^{(2,1)}\Delta^{(1^{3})}(s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)})$ is not zero because the summand $s_{(3n+|\alpha|)}$ in the Schur function expansion of $s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)}$ is not cancelled. In order to show that the term $\Delta^{(3,3)}_{2}\Delta^{(3)}\Delta^{(1^{3})}(s_{(n+\alpha_{1})}s_{(n+\alpha_{2})}s_{(n)})$ is not zero we show that the multiplicity of $s_{(2n+|\alpha|,n)}$ in its Schur function decomposition is not zero. We denote the multiplicity of a Schur function $s_{\lambda}$ in a symmetric function $f$ by ${\mathrm{mult}}(\lambda,f)$. Now, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The four involved numbers count in this order the number of matrices in $\Delta^{(1^{4})}P_{4,n,\alpha}^{{\mathbb{Z}}}$ of the form

[TABLE]

[TABLE]

[TABLE]

[TABLE]

It follows

[TABLE]

[TABLE]

$\blacksquare$

The next lemmas are dedicated to the statement Conjecture 2.0 about fourfold products.

Lemma 3.4**.**

We have

[TABLE]

[TABLE]

Proof.

Lemma 3.3 yields

[TABLE]

There is an injective map $\Delta^{(2,1,1)}\Delta^{(1^{4})}P_{4,n-1,\alpha}^{{\mathbb{Z}}}\rightarrow\Delta^{(2,1,1)}\Delta^{(1^{4})}P_{4,n,\alpha}^{{\mathbb{Z}}}$ given by

[TABLE]

The matrices that are not hit are those with $a_{13}=0\lor a_{21}=0\lor a_{32}=0\lor a_{31}=a_{21}\lor a_{21}+a_{22}+a_{23}=a_{11}+a_{12}+a_{13}$. $a_{21}=0$ implies $a_{31}=a_{21}$ because of $0\leq a_{31}\leq a_{21}$. $\blacksquare$

Let $(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}:=\{A\in\Delta^{(2,1,1)}\Delta^{(1^{4})}P_{4,n,\alpha}:~{}a_{13}=0\lor a_{32}=0\lor a_{31}=a_{21}\lor a_{21}+a_{22}+a_{23}=a_{11}+a_{12}+a_{13}\}$. We know so far that

[TABLE]

is a sequence of unions of faces of $P_{4,n,\alpha}$ with corresponding dimensions

[TABLE]

where we say that the dimension of a union of polytopes is the maximum of the dimensions of the polytopes. Here, we have again that the subset $(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}\subseteq\Delta^{(2,1,1)}\Delta^{(1^{4})}P_{4,n,\alpha}$ is a union of faces. Its dimension is $3$ because the face $\{A\in P_{4,n,\alpha}:~{}a_{41}=a_{14}=a_{13}=0\}\subseteq(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}$ contains the $4$ affine independent points

[TABLE]

[TABLE]

Lemma 3.5**.**

We have

[TABLE]

[TABLE]

Proof.

There is an injection $(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n-1,\alpha}^{{\mathbb{Z}}}\rightarrow(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}^{{\mathbb{Z}}}$ given by

[TABLE]

The matrices that are not hit are those with $a_{12}=0$ or $a_{22}=0$ or $a_{23}=0$ or $a_{21}+a_{22}=a_{11}+a_{12}$ or $a_{31}+a_{32}=a_{21}+a_{22}$. $\blacksquare$

Let $\Delta^{(2,2)}(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}:=\{A\in(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}:~{}a_{12}=0\lor a_{22}=0\lor a_{23}=0\lor a_{21}+a_{22}=a_{11}+a_{12}\lor a_{31}+a_{32}=a_{21}+a_{22}\}$. The face $\{A\in P_{4,n,\alpha}:~{}a_{41}=a_{14}=a_{13}=a_{23}=0\}\subseteq\Delta^{(2,2)}(\Delta^{(2,1,1)})^{2}\Delta^{(1^{4})}P_{4,n,\alpha}$ has dimension $2$ because it contains the $3$ affine independent points

[TABLE]

[TABLE]

We summarize that

[TABLE]

is a sequence of unions of faces of $P_{4,n,\alpha}$ with corresponding dimensions

[TABLE]