Homomorphisms between Specht modules

Abstract

In positive characteristic, the Specht modules S λ corresponding to partitions λ are not necessarily irreducible, and understanding their structure is vital to understanding the representation theory of the symmetric group. In this paper, we address the related problem of finding the spaces of homomorphisms between Specht modules. Indeed in [2], Carter and Payne showed that the space of homomorphisms from S λ to S µ is non-zero for certain pairs of partitions λ and µ where the Young diagram for µ is obtained from that for λ by moving several nodes from one row to another. We also consider partitions of this type, and, by explicitly examining certain combinations of semi-standard homomorphisms, we are able to give a constructive proof of the Carter-Payne theorem and to generalise it.

Homomorphisms and Higher Extensions for Schur Algebras and Symmetric Groups

“…The following result is due to Fayers and Martin [FM04,Theorem 22], and (although this is not obvious) generalises Theorem 4.2.…”

“…The methods used to date in such calculations provide no compelling indication either way. As we will see in Remark 5.4(iii), a potential source of large Hom-spaces has been given by Fayers and Martin [FM04]; however no actual examples have yet been found.…”

“…This is proved first for SL n by looking at the action of the hyperalgebra on Weyl modules, and then by applying Theorem 3.1 to get the result for the symmetric group. It has recently been generalised by Fayers and Martin [FM04], who work in the symmetric group setting by looking at semi-standard homomorphisms. We will return to this generalisation in Section 5.…”

“…(i) The only new example given in [FM04] as an application of their main theorem is the case λ = (7, 3) and µ = (4, 3…”

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

“…In the special cases we need to consider, the result can be expressed in a very simple way as a multiple of another semi-standard homomorphism. The lemmas in this section can be obtained by applying Lemmas 5 and 7 from the paper [4] of Fayers and Martin. We include a direct proof of Lemma 5.5, because we need this proof in Section 7.…”