Composition Multiplicities of Tensor Products and Branching Multiplicities for $\text{GL}(n)$

TL;DR

This paper investigates the relationship between composition multiplicities in tensor products of simple modules and branching multiplicities for GL(n), focusing on the wedge square of the dual natural module, extending prior work on fundamental modules.

Contribution

It introduces a method to determine composition multiplicities for tensor products involving the wedge square of the dual natural module for GL(n).

Findings

Established a link between tensor product multiplicities and branching multiplicities.

Provided a new computational approach for specific composition factors.

Extended previous results from fundamental modules to the wedge square of the dual natural module.

Abstract

Following work of Brundan and Kleshchev (2000), which considered tensor products with the natural module (and its dual) for $\text{GL}(n)$, we take the next fundamental module and explore the relationship between multiplicities of composition factors of tensor products of simple modules for $\text{GL}(n)$ and branching multiplicities. In doing this we provide a method for determining the composition multiplicities for some composition factors of tensor products of simple modules with the wedge square of the dual natural module.