On homomorphisms between Weyl modules: The case of a column transposition

Let $G=G{L}_n(K)$ be the general linear group defined over an infinite field $K$ of positive characteristic $p$ and let $\mathrm{\Delta }(\lambda )$ be the Weyl module of $G$ which corresponds to a partition $\lambda$. In this paper we classify all homomorphisms $\mathrm{\Delta }(\lambda )\to \mathrm{\Delta }(\mu )$ when $\lambda =(a,b,1^d)$ and $\mu =(a+d,b)$, $d>1$. In particular, we show that $Ho{m}_G(\mathrm{\Delta }(\lambda ),\mathrm{\Delta }(\mu ))$ is nonzero if and only if $p=2$ and $a$ is even. In this case, we show that the dimension of the homomorphism space is equal to 1 and we provide an explicit generator whose description depends on binary expansions of various integers. We also show that these generators in general are not compositions of Carter-Payne homomorphisms.