Decomposable Specht modules indexed by bihooks II

TL;DR

This paper investigates the structure of decomposable Specht modules labeled by bihooks over the Iwahori--Hecke algebra of type B, revealing their semisimplicity and exact composition factors, and determining their graded decomposition numbers.

Contribution

It characterizes when these Specht modules are semisimple or nearly so, and provides their detailed structure and decomposition numbers, advancing understanding of their decomposition behavior.

Findings

Many decomposable Specht modules are semisimple or close to semisimple.

The exact structure and composition factors of these modules are determined.

Graded decomposition numbers are computed for almost all such modules.

Abstract

Previously, the last two authors found large families of decomposable Specht modules labelled by bihooks, over the Iwahori--Hecke algebra of type $B$. In most cases we conjectured that these were the only decomposable Specht modules labelled by bihooks, proving it in some instances. Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we look back at our decomposable Specht modules and show that they are often either semisimple, or very close to being so. We obtain their exact structure and composition factors in these cases. In the process, we determine the graded decomposition numbers for almost all of the decomposable Specht modules indexed by bihooks.