A note on the fusion product decomposition of Demazure modules

TL;DR

This paper completes the fusion product decomposition theorem for higher-level affine Demazure modules in specific exceptional types, providing a new combinatorial and case-free proof for the key fact involved.

Contribution

It extends the fusion product decomposition theorem to additional affine types and introduces a new uniform combinatorial proof for the core fact.

Findings

Decomposition theorem proven for $E^{(1)}_{6, 7, 8}$, $F^{(1)}_4$, and $E^{(2)}_{6}$.

New combinatorial proof for the key fact used in previous theorems.

Provides a case-free, uniform proof for the core combinatorial fact.

Abstract

We settle the fusion product decomposition theorem for higher-level affine Demazure modules for the cases $E^{(1)}_{6, 7, 8}, F^{(1)}_4$ and $E^{(2)}_{6}$, thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact, that was used in Chari et al. (op cit.), to prove this decomposition theorem. We give a case free uniform proof for this key fact.