On homomorphisms indexed by semistandard tableaux

TL;DR

This paper investigates the structure of homomorphism spaces between Specht modules for Hecke algebras of type A, introducing new theorems and algorithms for explicit computation, especially for specific partition pairs.

Contribution

It presents a cellular analogue of the kernel intersection theorem, a $q$-analogue of Fayers and Martin's theorem, and an algorithm for computing homomorphism spaces between Specht modules.

Findings

Established a cellular analogue of the kernel intersection theorem.

Developed a $q$-analogue of a theorem by Fayers and Martin.

Provided an explicit description of homomorphism spaces for certain partitions.

Abstract

We study the homomorphism spaces between Specht modules for the Hecke algebras $\h$ of type $A$. We prove a cellular analogue of the kernel intersection theorem and a $q$-analogue of a theorem of Fayers and Martin and apply these results to give an algorithm which computes the homomorphism spaces $\Hom_{\h}(S^\mu,S^\lambda)$ for certain pairs of partitions $\lambda$ and $\mu$. We give an explicit description of the homomorphism spaces $\Hom_\h(S^\mu,S^\lambda)$ where $\h$ is an algebra over the complex numbers, $\lambda=(\lambda_1,\lambda_2)$ and $\mu$ is an arbitrary partition with $\mu_1 \geq \lambda_2$.