Filtration of tensor product of local Weyl modules for $\mathfrak{sl}_{n+1}[t]$

TL;DR

This paper studies the structure of tensor products of local Weyl modules for rak{sl}_{n+1}[t], revealing their filtrations and establishing isomorphisms with fusion products, advancing understanding of module decompositions.

Contribution

It provides a detailed analysis of filtrations of tensor products of local Weyl modules and proves isomorphisms with fusion products for rak{sl}_3[t], showing parameter independence.

Findings

Tensor products admit filtrations with quotients as truncated Weyl modules or fusion products.

Truncated Weyl modules for rak{sl}_3[t] are isomorphic to fusion products of irreducible modules.

Fusion product modules are independent of evaluation parameters.

Abstract

In this paper, we consider the tensor product of local Weyl modules for $\mathfrak{sl}_{n+1}[t]$ whose highest weights are multiples of the first and $n^{th}$ fundamental weights. We determine the graded character of these tensor product modules in terms of the graded character of local Weyl modules and prove that these modules admit a filtration whose successive quotients are either truncated Weyl modules or fusion products of Demazure modules. Furthermore, we establish that the truncated Weyl modules appearing as quotients in the filtration of tensor products of local Weyl modules of $\mathfrak{sl}_3[t]$ are indeed isomorphic to fusion products of irreducible $\mathfrak{sl}_3[t]$-modules which establish the independence of a family of fusion product modules of $\mathfrak{sl}_3[t]$ from the set of its evaluation parameters.