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

TL;DR

This paper provides a simple description of Carter-Payne homomorphisms between Specht modules using Murphy-Jucys elements, proves their one-dimensionality, and explores their implications for endomorphism rings and branching rules.

Contribution

It introduces a straightforward method to describe Carter-Payne homomorphisms and establishes their 1-dimensionality, with applications to endomorphism rings of Specht modules.

Findings

Homomorphisms described as maps between polytabloids

Hom space is proven to be 1-dimensional

Endomorphism ring is a direct product of truncated polynomial rings

Abstract

Let n be a positive integer and let p be a prime. Suppose that we take a partition of n, and obtain another partition by moving a node from one row to a shorther row. Carter and Payne showed that if the p-residue of the removed and added positions is the same, then there is a non-zero homomorphism between the corresponding Specht modules for the symmetric group of degree n, defined over a field of characteristic p. In this paper we give a very simple description of such a homomorphism, as a map between polytabloids, using the action of a Murphy-Jucys element. We also present a proof that in this context the homomorphism space is 1-dimensional. S. Lyle has already proved the more general result for Iwahori-Hecke algebras. In the process we give a formula for the Carter-Payne homomorphism as a linear combination of semi-standard homomorphisms. Our methods allow us to compute a lower bound for where the image of this homomorphism lies in the Jantzen filtration of the codomain Specht module. As an application, we show that the endomorphism ring of the restriction of a Specht module to the symmetric group of degree n-1 is an explicit direct product of truncated polynomial rings. A. Kleshchev proved the analogous result for the restriction of irreducible modules.