2. Preliminaries

2.1. Notation

Throughout this paper, K will be an infinite field of characteristic $p>0$ . We will be working with homogeneous polynomial representations of $GL_n(K)$ of degree r, or equivalently, with modules over the Schur algebra $S=S_K(n,r)$ . A standard reference here is [Reference Green8].

In what follows we fix notation and recall from Akin and Buchsbaum [Reference Akin and Buchsbaum1], and also Akin et al. [Reference Akin, Buchsbaum and Weyman2] important facts.

Let $V=K^n$ be the natural $GL_n(K)$ -module. The divided power algebra $DV=\sum_{i\geq 0}D_iV$ of V is defined as the graded dual of the Hopf algebra $S(V^{*})$ , where $V^{*}$ is the linear dual of V and $S(V^{*})$ is the symmetric algebra of $V^{*}$ , see [Reference Akin, Buchsbaum and Weyman2], I.4. For $v \in V$ and i, j nonnegative integers, we will use many times relations of the form

\[v^{(i)}v^{(j)}=\tbinom{i+j}{j}v^{(i+j)},\]

where $\tbinom{i+j}{j}$ is the indicated binomial coefficient.

By $\wedge(n,r)$ , we denote the set of sequences $a=(a_1, \dots, a_n)$ of nonnegative integers that sum to r and by $\wedge^+(n,r)$ we denote the subset of $\wedge(n,r)$ consisting of sequences $\lambda=(\lambda_1, \dots, \lambda_n)$ such that $\lambda_1 \ge \lambda_2 \dots \ge \lambda_n$ . Elements of $\wedge^+(n,r)$ are referred to as partitions of r with at most n parts. The transpose partition $\lambda^t =(\lambda_1^t,...,\lambda_q^t) \in \wedge^+(\lambda_1,r), q=\lambda_1,$ of a partition $\lambda=(\lambda_1,...,\lambda_n) \in \wedge^+(n,r)$ is defined by $\lambda_j^t = \# \{i\,{:}\, \lambda_i \ge j\}$ .

If $a=(a_1,\dots, a_n) \in \wedge(n,r)$ , we denote by D(a) or $D(a_1,\dots,a_n)$ the tensor product $D_{a_1}V\otimes \dots \otimes D_{a_n}V$ . All tensor products in this paper are over K.

The exterior algebra of V is denoted $\Lambda V=\sum_{i\geq 0}\Lambda^iV$ . If $a=(a_1,\dots, a_n) \in \wedge(n,r)$ , we denote by $\Lambda(a)$ the tensor product $\Lambda^{a_1}V\otimes \dots \otimes \Lambda^{a_n}V$ .

For $\lambda \in \wedge^+(n,r)$ , we denote by $\Delta(\lambda)$ the corresponding Weyl module for S. In [Reference Akin, Buchsbaum and Weyman2], Definition II.1.4, the module $\Delta(\lambda)$ (denoted $K_{\lambda}F$ there), was defined as the image a particular map $d^{\prime}_{\lambda} \,{:}\, D(\lambda) \to \Lambda(\lambda^{t})$ . For example, if $\lambda =(r)$ , then $\Delta(\lambda) =D_rV$ , and if $\lambda =(1^r)$ , then $\Delta(\lambda) =\Lambda^{r}V$ .

2.2. Relations for Weyl modules.

We recall from [Reference Akin, Buchsbaum and Weyman2], Theorem II.3.16, the following description of $\Delta (\lambda)$ in terms of generators and relations.

Theorem 2.1 ([Reference Akin, Buchsbaum and Weyman2]). Let $\lambda=(\lambda_1,\dots,\lambda_m) \in \wedge^+(n,r)$ , where $\lambda_m >0$ . There is an exact sequence of S-modules

\[ \sum_{i=1}^{m-1}\sum_{t=1}^{\lambda_{i+1}}D(\lambda_1,\dots,\lambda_i+t,\lambda_{i+1}-t,\dots,\lambda_m) \xrightarrow{\square_{\lambda}} D(\lambda) \xrightarrow{d^{\prime}_\lambda} \Delta(\lambda) \to 0, \]

where the restriction of $ \square_{\lambda} $ to the summand $M_i(t)=D(\lambda_1,\dots,\lambda_i+t,\lambda_{i+1}-t,\dots,\lambda_m)$ is the composition

\[ M_i(t) \xrightarrow{1\otimes\cdots \otimes \Delta \otimes \cdots 1}D(\lambda_1,\dots,\lambda_i,t,\lambda_{i+1}-t,\dots,\lambda_m)\xrightarrow{1\otimes\cdots \otimes \eta \otimes \cdots 1} D(\lambda), \]

where $\Delta\,{:}\,D(\lambda_i+t) \to D(\lambda_i,t)$ and $\eta\,{:}\,D(t,\lambda_{i+1}-t) \to D(\lambda_{i+1})$ are the indicated components of the comultiplication and multiplication, respectively, of the Hopf algebra DV and $d^{\prime}_\lambda$ is the map in [Reference Akin, Buchsbaum and Weyman2], Def.II.13.

2.3. Standard basis of $\Delta(\mu)$

We will record here and in the next subsection two important facts from [Reference Akin, Buchsbaum and Weyman2] and [Reference Akin and Buchsbaum1] specified to the case of partitions consisting of two parts.

Let us fix the order $e_1<e_2<...<e_n$ on the set $\{e_1, e_2, ..., e_n\}$ of the canonical basis elements of the natural module V of $GL_n(K)$ . We will denote each element $e_i$ by its subscript i. For a partition $\mu=(\mu_1,\mu_2) \in \wedge^+(n,r)$ , a tableau of shape $\mu$ is a filling of the diagram of $\mu$ with entries from $\{1,...,n\}$ . Such a tableau is called standard if the entries are weakly increasing across the rows from left to right and strictly increasing in the columns from top to bottom. (The terminology used in [Reference Akin, Buchsbaum and Weyman2] is ’co-standard’).

The set of standard tableaux of shape $\mu$ will be denoted by $\textrm{ST}(\mu)$ . The weight of a tableau T is the tuple $\alpha=(\alpha_1,...,\alpha_n)$ , where $\alpha_i$ is the number of appearances of the entry i in T. The subset of $\textrm{ST}(\mu)$ consisting of the (standard) tableaux of weight $\alpha$ will be denoted by $\textrm{ST}_{\alpha}(\mu).$

For example, the following tableau of shape $\mu=(6,4)$ .

is standard and has weight $\alpha=(3,4,1,2)$ .

We will use ‘exponential’ notation for standard tableaux. Thus for the above example, we write

\[ T=\begin{array}{l} 1^{(3)}2^{(2)}4 \\ \\[-8pt] 2^{(2)}34 \end{array}. \]

To each tableau T of shape $\mu=(\mu_1, \mu_2)$ , we may associate an element

\[x_T=x_T(1) \otimes x_T(2) \in D(\mu_1,\mu_2),\]

where $x_T(i)=1^{(a_{i1})}\cdots n^{(a_{in})}$ and $a_{ij}$ is equal to the number of appearances of j in the i-th row of T. For example, the T depicted above yields $x_T=1^{(3)}2^{(2)}4\otimes2^{(2)}34$ . According to [Reference Akin, Buchsbaum and Weyman2], Theorem II.2.16, we have the following.

Theorem 2.2 ([Reference Akin, Buchsbaum and Weyman2]). The set $\{d^{\prime}_{\mu}(x_T)\,{:}\, T \in \textrm{ST}(\mu)\}$ is a basis of the K-vector space $\Delta({\mu})$ .

If $x=1^{(a_1)}2^{(a_2)}\cdots n^{(a_n)} \otimes 1^{(b_1)}2^{(b_2)}\cdots n^{(b_n)} \in D(\mu)$ , we will denote the element $d^{\prime}_{\mu}(x) \in \Delta(\mu)$ by

\[ \left[\begin{array}{l} 1^{(a_1)} 2^{(a_2)} \cdots n^{(a_n)} \\ \\[-8pt] 1^{(b_1)} 2^{(a_2)} \cdots n^{(b_n)} \end{array}\right]. \]

2.4. Weight subspaces of $\Delta(\mu)$

Suppose $n \ge r$ . Let $\nu \in \wedge(n,r)$ and $\mu=(\mu_1,\mu_2) \in \wedge^+(2,r)$ . According to [Reference Akin and Buchsbaum1], equation (11), a basis of the K-vector space $\textrm{Hom}_S(D(\nu), \Delta(\mu))$ is in 1-1 correspondence with set $\textrm{ST}_{\nu}(\mu)$ of standard tableaux of shape $\mu$ and weight $\nu$ .

For the computations to follow, we need to make the above correspondence explicit. Let $\nu = (\nu_1,..., \nu_n) \in \wedge(n,r)$ and $T \in \textrm{ST}_{\nu}(\mu)$ . Let $a_i$ (respectively, $b_i$ ) be the number of appearances of i in the first row (respectively, second row) of T. We note that $\nu_i = a_i+b_i$ for each i. In particular, we have $a_1 = \nu_1$ because of standardness of T. Define the map

\[\phi_T\,{:}\,D(\nu) \to \Delta(\mu),\]

\[ x_1 \otimes x_2 \otimes \cdots\otimes x_n \mapsto \sum_{i_2,...,i_n}d^{\prime}_{\mu}\!\left(x_1x_{2i_2}(a_2)\cdots x_{ni_n}\!(a_n) \otimes x_{2i_2}(b_2)'\cdots x_{ni_n}(b_n)'\right), \]

where $\sum_{i_s}x_{si_s}(a_s) \otimes x_{si_s}(b_s)'$ is the image of $x_s$ under the component

\[D(\nu_s) \to D(a_s,b_s),\]

of the diagonalization $\Delta \,{:}\, DV \to DV \otimes DV$ of the Hopf algebra DV for $s=2,...,n$ . Thus we have that a basis of the K-vector space $\textrm{Hom}_S(D(\nu), \Delta(\mu))$ is the set

\[\{\phi_T\,{:}\, T \in \textrm{ST}_{\nu}(\mu)\}.\]

In particular, suppose $\lambda=(\lambda_1,...,\lambda_m) \in \wedge^+(n,r)$ is a partition and $\mu =(\mu_1,\mu_2) \in \wedge^+(2,r)$ satisfies $\mu_2 \le \lambda_1$ . This inequality means that each tableau of shape $\mu$ that has the form

\[ \begin{array}{l} 1^{(\lambda_1)} 2^{(a_2)} \cdots m^{(a_m)} \\ 2^{(b_2)} \cdots m^{(b_m)}\end{array}\]

is standard. Hence, we have the following result.

Lemma 2.3. Suppose $n \ge r$ . Let $\lambda,\mu \in \wedge^{+}(n,r)$ , where $\lambda=(\lambda_1,...,\lambda_m)$ and $ \mu=(\mu_1, \mu_2)$ . If $\mu_2 \le \lambda_1$ , than a basis of the K- vector space $\textrm{Hom}_S(D(\lambda), \Delta(\mu))$ is given by the elements $\phi_T$ , where

\[T= \begin{array}{l} 1^{(\lambda_1)} 2^{(a_2)} \cdots m^{(a_m)} \\ 2^{(b_2)} \cdots m^{(b_m)} \end{array},\]

are such that

\begin{align*} &a_i, b_i \ge 0, \; a_i+b_i=\lambda_i, (i=2,...,m),\\[3pt] &a_2+\cdots+a_m=\mu_1-\lambda_1, \; b_2+\cdots+b_m=\mu_2. \end{align*}

Example. Suppose $\lambda=(\lambda_1,3,3)$ and $\mu=(\lambda_1+4,2)$ , where $\lambda_1 \ge 3$ . Then $\{[T_1],[T_2],[T_3]\}$ is a basis of $\textrm{Hom}_S(D(\lambda), \Delta(\mu))$ , where

\[T_1= \begin{array}{l} 1^{(\lambda_1)} 2^{(3)}3 \\ 3^{(2)} \end{array}, T_2= \begin{array}{l} 1^{(\lambda_1)} 2^{(2)}3^{(2)} \\ 23\end{array}, T_3= \begin{array}{l}1^{(\lambda_1)} 23^{(3)} \\2^{(2)}\end{array}.\]

For $x=1^{(\lambda_1)} \otimes 1^{(2)}2 \otimes 3^{(3)} \in D(\lambda)$ and $T=T_2$ , we have

\[\phi_T(x)=\binom{\lambda_1+2}{2}\left[\begin{array}{l} 1^{(\lambda_1+2)} 3^{(2)} \\ \\[-9pt] 23\end{array}\right]+\binom{\lambda_1+1}{1}\left[\begin{array}{l} 1^{(\lambda_1+1)} 23^{(2)} \\ \\[-9pt] 13\end{array}\right],\]

where the binomial coefficients come from multiplication in the divided power algebra DV.

3. Main result

In order to state the main result of this paper, we use the following notation. If x, y are positive integers, let

\[R(x,y)=\gcd\!\left\{\binom{x}{1}, \binom{x+1}{2},...,\binom{x+y-1}{y}\right\}.\]

If x is a positive integer, let $R(x,0)=0$ .

Theorem 3.1. Let K be an infinite field of characteristic $p>0$ and let $n \ge r$ be positive integers. Let $\lambda,\mu \in \wedge^{+}(n,r)$ be partitions such that $\lambda=(\lambda_1,...,\lambda_m)$ and $\mu =(\mu_1, \mu_2)$ , where $\lambda_m \neq 0$ , $m \ge 2$ and $\mu_2 \le \lambda_1 \le \mu_1$ . If p divides all of the following integers

\begin{align*} &R(\lambda_1 -\mu_2+1,l), l=\min\{\lambda_2, \mu_1 -\lambda_1\} \\[3pt] &R(\lambda_i+1,\lambda_{i+1}), i=2,...,m-1, \end{align*}

then the map

\[\psi =\sum_{T \in \textrm{ST}_{\lambda}(\mu)}\phi_T\]

induces a nonzero homomorphism $\Delta(\lambda) \to \Delta(\mu).$

Remark 3.2. Consider the symmetric group $\mathfrak{S}_r$ on r symbols. For a partition $\lambda$ of r, let $\textrm{Sp}(\lambda)$ be the corresponding Specht module defined in Section 6.3 of [Reference Green8]. From Theorem 3.7 of [Reference Carter and Lusztig3], we have

\[\dim \textrm{Hom}_S(\Delta(\lambda),\Delta(\mu)) \le \dim \textrm{Hom}_{\mathfrak{S}_r}(\textrm{Sp}(\mu),\textrm{Sp}(\lambda)),\]

for all partitions $\lambda, \mu$ of r. (In fact we have equality if $p>2$ according to loc. cit.) Hence, our Theorem 3.1 may be considered as a nonvanishing result for homomorphisms between Specht modules.

For further use, we note that the divisibility assumptions of Theorem 3.1 may be stated in a different way. For a positive integer y, let $l_{p}(y)$ be the least integer i such that $p^i>y$ . From James [Reference James9], Corollary 22.5, we have the following result.

4. Straightening

For the proof of Theorem 3.1, we will need the following identities involving binomial coefficients. Our convention is that $\tbinom{a}{b}=0$ if $b>a$ or $b<0$ .

Proof.

1. (1) The identity in (a) is Vandermonde’s identity. For (b), we have

\begin{align*} \sum_{j_0+\cdots+j_s = m}(\!-1)^{j_0}\binom{m_1}{ j_1}\cdots \binom{m_s}{j_s}&= \sum_{j_0=0}^{m} \; \sum_{j_1+\cdots+j_s = m-j_0}(-\!1)^{j_0}\binom{m_1}{ j_1}\cdots \binom{m_s}{j_s}\\& =\sum_{j_0=0}^{m} (\!-1)^{j_0}\sum_{j_1+\cdots+j_s = m-j_0}\binom{m_1}{ j_1}\cdots \binom{m_s}{j_s}\\&= \sum_{j_0=0}^{m} (\!-1)^{j_0}\binom{m}{m-j_0}\\&= 0. \end{align*}

1. (2) The second identity is Lemma 2.6 of [Reference Lyle14] for $q=1$ . The first follows from the second with the substitution $ j \mapsto c-j$ .

We will also need the following explicit form of the straightening law concerning violations of standardness in the first column.

Proof.

1. (1) This is clear since there is no element in $\Delta(\mu)$ of weight $(\nu_1,...,\nu_n)$ satisfying $\nu_1 > \mu_1$ .

2. (2) We proceed by induction on $b_1$ , the case $b_1=0$ being clear. Suppose $b_1>0$ . Consider the element $x \in D(\mu_1+b_1, \mu_2-b_1)$ , where

   \[x=1^{(a_1+b_1)}2^{(a_2)}\cdots n^{(a_n)} \otimes 2^{(b_2)}\cdots n^{(b_n)},\]
   and the map
   \begin{align*} \delta \,{:}\, D(\mu_1+b_1, \mu_2-b_1) \xrightarrow{\Delta \otimes 1} D(\mu_1, b_1, \mu_2-b_1) \xrightarrow{1 \otimes \eta } D(\mu_1, \mu_2). \end{align*}
   According to the analogue of Lemma II.2.9 of [Reference Akin, Buchsbaum and Weyman2] for divided powers in place of exterior powers, we have $d^{\prime}_{\mu}(\delta(x))=0$ in $\Delta(\mu_1,\mu_2)$ . Thus

\begin{equation*} \left[\begin{array}{l} 1^{(a_1)} \cdots n^{(a_n)} \\ 1^{(b_1)} \cdots n^{(b_n)} \end{array}\right]=-\sum_{j_1,...,j_n} \binom{b_2+j_2}{b_2} \cdots \binom{b_n+j_n}{b_n}\left[\begin{array}{l} 1^{(a_1+b_1-j_1)} 2^{(a_2-j_2)} \cdots n^{(a_n-j_n)} \\ 1^{(j_1)}2^{(b_2+j_2)} \cdots n^{(b_n+j_n)} \end{array}\right], \end{equation*}

where the sum ranges over all nonnegative integers $j_1, ..., j_n$ such that $j_1+\cdots+j_n=b_1$ , $j_1<b_1$ and $j_s \le a_s$ for all $s=2,...,n$ . Let X be the right hand side of the above equality. By induction, we have

\begin{align*} X=-\sum_{j_1,...,j_n} \binom{b_2+j_2}{b_2} \cdots \binom{b_n+j_n}{b_n}(\!-1)^{j_1}\sum_{k_2,...,k_n} \binom{b_2+j_2+k_2}{b_2+j_2} \cdots \binom{b_n+j_n+k_n}{b_n+j_n} \\ \left[\begin{array}{l} 1^{(a_1+b_1)} 2^{(a_2-j_2-k_2)} \cdots n^{(a_n-j_n-k_n)} \\ 2^{(b_2+j_2+k_2)} \cdots n^{(b_n+j_n+k_n)} \end{array}\right], \end{align*}

where the new sum ranges over all nonnegative integers $k_2, ..., k_n$ such that $k_2+\cdots+k_n=j_1$ and $k_s \le a_s-j_s$ for all $s=2,...,n$ . Using the identities

\[\binom{b_s+j_s}{b_s} \binom{b_s+j_s+k_s}{b_s+j_s} =\binom{b_s+j_s+k_s}{b_s} \binom{j_s+k_s}{j_s}, \]

for $s=2,...,n$ , we obtain

\begin{align*} X=-\sum_{j_1,...,j_n,k_2,...,k_n}(\!-1)^{j_1} \binom{b_2+j_2+k_2}{b_2} \cdots \binom{b_n+j_n+k_n}{b_n}\binom{j_2+k_2}{j_2} \cdots \binom{j_n+k_n}{j_n} \\ \left[\begin{array}{l} 1^{(a_1+b_1)} 2^{(a_2-j_2-k_2)} \cdots n^{(a_n-j_n-k_n)} \\ 2^{(b_2+j_2+k_2)} \cdots n^{(b_n+j_n+k_n)} \end{array}\right]. \end{align*}

The coefficient c of

\[\left[\begin{array}{l} 1^{(a_1+b_1)} 2^{(a_2-i_2)} \cdots n^{(a_n-i_n)} \\ \\[-8pt] 2^{(b_2+i_2)} \cdots n^{(b_n+i_n)} \end{array}\right] \]

in the right hand side of the above equation is equal to

\[-\sum_{\substack{j_1,...,j_n,k_2,...,k_n \\ j_s+k_s=i_s}}(\!-1)^{j_1} \binom{b_2+j_2+k_2}{b_2} \cdots \binom{b_n+j_n+k_n}{b_n}\binom{j_2+k_2}{j_2} \cdots \binom{j_n+k_n}{j_n},\]

where the sum is restricted over those $j_1,...,j_n$ and $k_2,...,k_n$ that satisfy the additional conditions $j_s+k_s=i_s$ for all $s=2,...,n.$ Hence

\begin{align*}c&=-\sum_{j_1,...,j_n}(\!-1)^{j_1} \binom{b_2+i_2}{b_2} \cdots \binom{b_n+i_n}{b_n}\binom{i_2}{j_2} \cdots \binom{i_n}{j_n} \\& =-\binom{b_2+i_2}{b_2} \cdots \binom{b_n+i_n}{b_n}\sum_{j_1,...,j_n}(\!-1)^{j_1} \binom{i_2}{j_2} \cdots \binom{i_n}{j_n}.\\& \end{align*}

Remembering that in the last sum we have $j_1<b_1$ , Lemma 4.1 (1)(b) yields

\[\sum_{j_1,...,j_n}(\!-1)^{j_1} \binom{i_2}{j_2} \cdots \binom{i_n}{j_n}=0-(\!-1)^{b_1}.\]

Thus $c=(\!-1)^{b_1}\binom{b_2+i_2}{b_2} \cdots \binom{b_n+i_n}{b_n}.$

5. Proof of the main theorem

Consider the map $\psi \in \textrm{Hom}_S(D(\lambda), \Delta(\mu))$ given by the sum

\[\psi = \sum_{T \in \textrm{ST}_{\lambda}( \mu)}\phi_T,\]

in the statement of Theorem 3.1 We will show, according to Theorem 2.1, that $\psi(x)=0$ for every $x \in Im(\square_\lambda)$ . First we look at the relations corresponding to rows 1 and 2 of $\Delta(\lambda)$ .

Relations from rows 1 and 2

Let $x=1^{(\lambda_1)}\otimes 1^{(t)}2^{(\lambda_2-t)}\otimes3^{(\lambda_3)}\cdots m^{(\lambda_m)} \in Im(\square_\lambda),$ where $t\le \lambda_2$ , and let $T \in \textrm{ST}_{\lambda}(\mu).$ Then T is of the form

\[T=\begin{array}{l} 1^{(\lambda_1)}2^{(a_2)} \cdots m^{(a_m)} \\ 2^{(b_2)} \cdots m^{(b_m)} \end{array} \in \textrm{ST}_{\lambda}(\mu),\]

where the $a_i, b_i$ satisfy the conditions of Lemma 2.3. Using the definition of $\phi_T$ from 2.4, we have

\begin{align*}\phi_T(x)= \sum_{i\le t}\binom{\lambda_1+i}{i}\left[\begin{array}{l} 1^{(\lambda_1+i)}2^{(a_2-i)}3^{(a_3)} \cdots m^{(a_m)} \\ 1^{(t-i)}2^{(\lambda_2-t-a_2+i)}3^{(b_3)} \cdots m^{(b_m)} \end{array}\right].\end{align*}

If $(\lambda_1+i)+(t-i) \ge \mu_1$ , then by the first part of Lemma 4.2 we obtain $\phi_T(x)=0$ . Hence we may assume that $t \le min\{\lambda_2, \mu_1-\lambda_1\}. $ Using the second part of Lemma 4.2, we have

\begin{align*}\phi_T(x)=& \sum_{i\le t}\binom{\lambda_1+i}{i}(\!-1)^{t-i} \sum_{k_2+\cdots+k_m=t-i}\binom{b_2-k_3-\cdots-k_m}{k_2}\binom{b_3+k_3}{k_3}\cdots\binom{b_m+k_m}{k_m}\\&\qquad \left[\begin{array}{l} 1^{(\lambda_1+t)}2^{(a_2+k_3+\cdots+k_m)}3^{(a_3-k_3)} \cdots m^{(a_m-k_m)} \\ 2^{(b_2-k_3-\cdots-k_m)}3^{(b_3+k_3)} \cdots m^{(b_m+k_m)} \end{array}\right].\end{align*}

Let $c \in K$ be the coefficient of $\left[\begin{array}{l} 1^{(\lambda_1+t)}2^{(a_2+k_3+\cdots+k_m)}3^{(a_3-k_3)} \cdots m^{(a_m-k_m)} \\ 2^{(b_2-k_3-\cdots-k_m)}3^{(b_3+k_3)} \cdots m^{(b_m+k_m)}\end{array}\right] $ in the right hand side of the last equation and let $k=k_3+\cdots+k_m$ . Then

\begin{align*}c&= \left(\sum_{i=0}^{t}\binom{\lambda_1+i}{i}(\!-1)^{t-i}\binom{b_2-k}{t-k-i}\right) \binom{b_3+k_3}{k_3}\cdots\binom{b_m+k_m}{k_m}\\&= (\!-1)^k\left(\sum_{i=0}^{t-k}\binom{\lambda_1+i}{i}(\!-1)^{t-k-i}\binom{b_2-k}{t-k-i}\right) \binom{b_3+k_3}{k_3}\cdots\binom{b_m+k_m}{k_m}\\&= (\!-1)^{k}\binom{\lambda_1-b_2+t}{t-k}\binom{b_3+k_3}{k_3}\cdots\binom{b_m+k_m}{k_m},\end{align*}

where in the third equality we used the first identity of Lemma 4.1 (2). Thus

\begin{align*}\phi_T(x)=& \sum_{k_3,...,k_m}(\!-1)^k\binom{\lambda_1-b_2+t}{t-k}\binom{b_3+k_3}{k_3}\cdots\binom{b_m+k_m}{k_m} \left[\begin{array}{l} 1^{(\lambda_1+t)}2^{(a_2+k)}3^{(a_3-k_3)} \cdots m^{(a_m-k_m)} \\ 2^{(b_2-k)}3^{(b_3+k_3)} \cdots m^{(b_m+k_m)} \end{array}\right],\end{align*}

where $k=k_3+\cdots+k_m$ and the sum ranges over all nonnegative integers $k_3,...,k_m$ such that $k \le b_2$ and $k_s \le a_s$ for all $s=3,...,m$ .

By summing with respect to $T \in \textrm{ST}_{\lambda}({\mu})$ and using Lemma 2.3, we obtain

(1) \begin{align}\psi(x)=&\sum_{b_2,...,b_m}\sum_{k_3,...,k_m}(\!-1)^k\binom{\lambda_1-b_2+t}{t-k}\binom{b_3+k_3}{k_3}\cdots\binom{b_m+k_m}{k_m}\end{align}

\begin{align}& \left[\begin{array}{l} 1^{(\lambda_1+t)}2^{(a_2+k)}3^{(a_3-k_3)} \cdots m^{(a_m-k_m)} \\ 2^{(b_2-k)}3^{(b_3+k_3)} \cdots m^{(b_m+k_m)} \end{array}\right]\nonumber,\end{align}

where the new sum is over all nonnegative integers $b_2,...,b_m$ such that $b_i \le \lambda_i (i=2,...,m)$ and $b_2+\cdots+b_m=\mu_2$ .

Fix

\[[S]=\left[\begin{array}{l} 1^{(\lambda_1+t)}2^{(a_2+k)}3^{(a_3-k_3)} \cdots m^{(a_m-k_m)} \\ 2^{(b_2-k)}3^{(b_3+k_3)} \cdots m^{(b_m+k_m)}\end{array}\right] \in \Delta(\mu),\]

in the right hand side of (1) and let $q=\mu_2-(b_3+k_3)-\cdots-(b_m+k_m).$ Then $q= b_2-k$ . The coefficient of [S] in  (1) is equal to

\begin{align*} &\sum_{k}(\!-1)^k\binom{\lambda_1-q-k+t}{t-k} \sum_{k_3+\cdots+k_m=k}\binom{b_3+k_3}{k_3} \cdots \binom{b_m+k_m}{k_m}\\ &=\sum_{k}(\!-1)^k\binom{\lambda_1-q-k+t}{t-k} \binom{\mu_2 -q}{k}\\& =\binom{\lambda_1-\mu_2+t}{t}\\& =0,\end{align*}

where in the first equality we used Lemma 4.1 (1)(a) and in the second equality we used the second identity of Lemma 4.1 (2).

Relations from rows i and $i+1$ ( $i>1$ ).

This computation is similar to the previous one but simpler as there is no straightening. Let $y=1^{(\lambda_1)}\otimes\cdots\otimes {i}^{(\lambda_i)} \otimes i^{(t)}(i+1)^{(\lambda_{i+1}-t)} \otimes\cdots\otimes m^{(\lambda_m)} \in Im(\square_\lambda),$ where $ i>1$ and $t \le \lambda_{i+1}$ . As before let

\[T=\begin{array}{l} 1^{(a_1)} \cdots m^{(a_m)} \\ 2^{(b_2)} \cdots m^{(b_m)} \end{array} \in \textrm{ST}_{\lambda}(\mu).\]

The definition of $\phi_T$ yields

\begin{align*}\phi_T(y)= \sum_{j\le t}\binom{a_i+j}{j}\binom{b_i+t-j}{t-j}\left[\begin{array}{l} 1^{(\lambda_1)}2^{(a_2)}\cdots i^{(a_i+j)}(i+1)^{(a_{i+1}-j)} \cdots m^{(a_m)} \\ 2^{(b_2)}\cdots i^{(b_i+t-j)} (i+1)^{(b_{i+1}-t+j)}\cdots m^{(b_m)} \end{array}\right].\end{align*}

By summing with respect to $T \in \textrm{ST}_{\lambda}(\mu)$ and using Lemma 2.3, we have

(2) \begin{align}\psi(y)= &\sum_{b_2,...,b_m}\sum_{j\le t}\binom{\lambda_i - b_i+j}{j}\binom{b_i+t-j}{t-j}\end{align}

\begin{align}&\left[\begin{array}{l} 1^{(\lambda_1)}2^{(\lambda_2-b_2)}\cdots i^{(\lambda_i-b_i+j)}(i+1)^{(\lambda_{i+1}-b_{i+1}-j)} \cdots m^{(\lambda_m-b_m)} \\ 2^{(b_2)}\cdots i^{(b_i+t-j)} (i+1)^{(b_{i+1}-t+j)}\cdots m^{(b_m)},\nonumber\end{array}\right]\end{align}

where the new sum ranges over all nonnegative integers $b_2,...,b_m$ such that $b_i \le \lambda_i \; (i=2,...,m)$ and $b_2+\cdots+b_m=\mu_2$ .

Fix

\[[S]=\left[\begin{array}{l} 1^{(\lambda_1)}2^{(\lambda_2-b_2)}\cdots i^{(\lambda_i-b_i+j)}(i+1)^{(\lambda_{i+1}-b_{i+1}-j)} \cdots m^{(\lambda_m-b_m)} \\ 2^{(b_2)}\cdots i^{(b_i+t-j)} (i+1)^{(b_{i+1}-t+j)}\cdots m^{(b_m)}\nonumber\end{array}\right] \in \Delta(\mu),\]

in the right hand side of (2) and let $q=b_i-j$ . The coefficient of [S] in (2) is equal to

\begin{align*}\sum_{j\le t}\binom{\lambda_j - q}{j}\binom{t+q}{t-j} =\binom{\lambda_i+t}{t}=0,\end{align*}

where in the first equality we used Lemma 4.1 (1)(a).

We have shown thus far that the map $\psi = \sum_{T \in \textrm{ST}(\lambda, \mu)}\phi_T$ induces a homomorphism of S-modules $\bar{\psi}\,{:}\, \Delta(\lambda) \to \Delta(\mu)$ and it remains to be shown that $\bar{\psi} \neq 0$ . Let $z=1^{(\lambda_1)}\otimes\cdots\otimes m^{(\lambda_m)} \in D(\lambda)$ and $T \in \textrm{ST}_{\lambda}(\mu).$ Then from the definition of $\phi_T$ we have $ \phi_T(x)=[T]$ and hence

\[\psi(x)=\sum_{T \in \textrm{ST}_{\lambda}(\mu)}[T].\]

The right hand side is a sum of distinct basis elements in $\Delta(\mu)$ (each with coefficient 1) according to Theorem 2.2 and hence nonzero. The proof is complete.

6. Homomorphism spaces of dimension greater than 1

As mentioned in Introduction, the first examples of Weyl modules $\Delta(\lambda),\Delta(\mu)$ such that $\dim\textrm{Hom}_S(\Delta(\lambda),\Delta(\mu))>1$ were obtained by Dodge [Reference Dodge6]. More examples were found by Lyle [Reference Lyle14], in fact in the q-Schur algebra setting. The purpose of this section is to observe that the homomorphism spaces of Theorem 3.1 may have dimension $>1$ , see Corollary 6.2 and Example 6.4 below.

We recall the following special case of the classical nonvanishing result of Carter and Payne [Reference Carter and Payne4]. Here, boxes are raised between consecutive rows. See [Reference Maliakas16], 1.2 Lemma, for a proof of this particular case in our context.

Proposition 6.1. ([Reference Carter and Payne4]). Let $n \ge r$ . Let $\lambda, \mu \in \wedge^+(n,r)$ such that for some some $d>0$ we have $\mu=(\lambda_1+d,\lambda_2-d,\lambda_3,...,\lambda_m)$ , where $\lambda=(\lambda_1,...,\lambda_m)$ . Suppose p divides $R(\lambda_1-\lambda_2+d+1,d)$ . Then the map

\begin{align*} \alpha \,{:}\, D(\lambda_1,\lambda_2,...,\lambda_m) &\xrightarrow{1 \otimes \Delta \otimes 1 } D(\lambda_1,d, \lambda_2-d,...,\lambda_m)\\& \xrightarrow{\eta \otimes 1} D(\lambda_1+d,\lambda_2-d,...,\lambda_m), \end{align*}

where $\Delta \,{:}\, D(\lambda_2) \to D(d,\lambda_2-d)$ is the indicated diagonalization and $\eta \,{:}\, D(\lambda_1, d) \to D(\lambda_1+d)$ and the indicated multiplication, induces a nonzero homomorphism $\Delta(\lambda) \to \Delta(\mu)$ . The main result of this section is the following.

Proof. By the first two divisibility conditions, the map

\[\psi_1 =\sum_{T \in \textrm{ST}_{\lambda}(\mu)}\phi_T \,{:}\, D(\lambda) \to D(\mu),\]

induces a nonzero homomorphism $\bar{\psi_1}\,{:}\, \Delta(\lambda) \to \Delta(\mu)$ according to Theorem 3.1.

Next consider the following maps

\begin{align*} \alpha \,{:}\, D(\lambda_1,\lambda_2,...,\lambda_m) \to D(\lambda_1+d,\lambda_2-d,...,\lambda_m), \end{align*}

as in Proposition 6.1 and

\begin{align*} \beta \,{:}\, D(\lambda_1+d, \lambda_2-d,...,\lambda_m) &\xrightarrow{1 \otimes \eta' } D(\lambda_1+d, \lambda_2-d+\lambda_3+\cdots+\lambda_m), \end{align*}

where $\eta' \,{:}\, D(\lambda_2-d,...,\lambda_m) \to D(\lambda_2-d+\lambda_3+\cdots +\lambda_m)$ is the indicated multiplication.

Under assumption (3), we have that $\alpha$ induces a nonzero map

\[\bar{\alpha}\,{:}\, \Delta(\lambda) \to D(\lambda_1+d,\lambda_2-d,...,\lambda_m),\]

according to Proposition 6.1

Under assumptions (2) and (4), we have that $\beta$ induces a nonzero map

\[\bar{\beta}\,{:}\, \Delta(\lambda_1+d, \lambda_2-d,...,\lambda_m) \to \Delta(\lambda_1+d, \lambda_2-d+\lambda_3+\cdots +\lambda_m),\]

according to Theorem 2.1

Consider the composition $\bar{\psi_2} =\bar{\beta}\bar{\alpha} \,{:}\, \Delta(\lambda) \to \Delta(\mu)$ depicted below, where Weyl modules are indicated by the diagrams of the corresponding partitions.

It remains to be shown that the homomorphisms $\bar\psi_1, \bar\psi_2$ are linearly independent. Let $z=d^{\prime}_{\lambda}(1^{(\lambda_1)}\otimes\cdots\otimes m^{(\lambda_m)}) \in \Delta(\lambda)$ . From the definitions of the maps, we have

\begin{equation*} \bar\psi_1(z)=\sum_{T \in \textrm{ST}_{\lambda}(\mu)}[T],\end{equation*}

and

\begin{equation*}\bar\psi_2(z)=\left[\begin{array}{l} 1^{(\lambda_1)}2^{(d)} \\ 2^{(\lambda_2-d)} \cdots m^{(\lambda_m)}\end{array}\right].\end{equation*}

It is clear that $\begin{array}{l} 1^{(\lambda_1)}2^{(d)} \\ 2^{(\lambda_2-d)} \cdots m^{(\lambda_m)}\end{array} \in \textrm{ST}_{\lambda}(\mu).$ Since $\lambda_3>0$ , the set $\textrm{ST}_{\lambda}(\mu)$ contains at least two elements. Hence, from the above equations and Theorem 2.2, it follows that the maps $\bar\psi_1, \bar\psi_2$ are linearly independent.

Example 6.3. Let p be a prime and a an integer such that $a \ge (p^2+1)(p-1)$ and

\[a \equiv p-2 \mod p^2.\]

Consider the following partitions

\begin{align*}&\lambda=(a, 2p-1, (p-1)^{p^2}), \\& \mu=(a+p, (p^2+1)(p-1)),\end{align*}

where $p-1$ appears $p^2$ times as a row in $\lambda$ . Using Lemma 3.4, it easily follows that the assumptions (1) - (4) of Corollary 6.2 are satisfied. For example, we have

\[\lambda_1-\mu_{2}+1 \equiv p-2-(p^2+1)(p-1)+1 \equiv 0 \mod p^2,\]

and hence by Lemma 3.3, $d=p$ divides $R(\lambda_1 - \mu_{2}+1,d)$ which is assumption (1). Thus, $\dim\textrm{Hom}_S(\Delta(\lambda), \Delta(\mu)) \ge 2.$ Footnote ¹

For $p=2$ , the least a that satisfies the above requirements is $a=8$ and thus we have the partitions $\lambda=(8,3,1,1,1,1), \mu=(10,5)$ . This pair appears in Example 4, Subsection 2.3, of Lyle’s paper [Reference Lyle15] which prompted us to consider Corollary 6.2 and in particular the composition $\bar{\psi_2} =\bar{\beta}\bar{\alpha} \,{:}\, \Delta(\lambda) \to \Delta(\mu)$ .