authors: 
Harald Ellers, Department of Mathematics, Allegheny College, PA, USA
John Murray, Department of Mathematics, National University of Ireland Maynooth, Ireland

title: Carter-Payne homomorphisms and branching rules for endomorphism rings of Specht modules}

subjclass: 20C20, 20C30

date: January 30, 2009

keywords: Specht Module, Carter-Payne Homomorphism, Jucys-Murphy Element, Jantzen Layers

abstract: 
Let $\Sigma_n$ be the symmetric group of degree $n$, and let $F$ be a field of characteristic 
$p\ne 2$. Suppose that $\lambda$ is a partition of\/ $n+1$, that $\alpha$ and $\beta$ are 
partitions of\/ $n$ that can be obtained by removing a node of the same residue from $\lambda$, 
and that $\alpha$ dominates $\beta$. Let $S^\alpha$ and $S^\beta$ be the Specht modules, defined 
over $F$, corresponding to $\alpha$, respectively $\beta$. We give a very simple description of 
a non-zero homomorphism $\theta:S^\alpha\rightarrow S^\beta$ and present a combinatorial proof 
of the fact that dim$\,{\rm Hom}_{F\Sigma_n}(S^\alpha,S^\beta) = 1$. As an application, we 
describe completely the structure of the ring ${\rm End}_{F\Sigma_n}({S^\lambda}\,
{\downarrow_{\Sigma_n}})$. Our methods furnish a lower bound for the Jantzen submodule of 
$S^\beta$ that contains the image of $\theta$.

status: submitted for publication