Tensor Product Decompositions, Limits in Excellent Filtrations, Affine Weyl Group Orbits, and Tableaux Counting

TL;DR

This work provides a comprehensive combinatorial description of outer multiplicities in tensor products of fundamental affine $A_n^{(1)}$ modules by linking Misra–Wilson tableaux with Demazure-flag multiplicities. The authors develop a finite, computable bridge between tableau counts and multipartition counts via affine Weyl group orbits, culminating in a stable limit formula for type $A_n$ that expresses multiplicities as sums of multipartition counts weighted by a quadratic function $f_{i,\xi}$. A key achievement is the explicit bijection between the Misra–Wilson tableau sets and a disjoint union of multipartition sets, enabling purely combinatorial computations of otherwise representation-theoretic quantities. The results extend known rank-$1$ identities to general rank and lay groundwork for potential generalizations to other types, offering a robust framework that connects tableaux, Demazure flags, and partition theory within affine Lie theory.

Abstract

We express the outer multiplicities in the tensor products of two fundamental simple modules for an affine Kac-Moody algebra of type $A$ in terms of counting certain sets of multipartitions by exploring the stabilizing limits of certain excellent filtrations. This extends for all ranks a previously obtained result by Jakelić and the second author for rank $1$. The same outer multiplicities were previously computed by Misra and Wilson in terms of counting certain sets of tableaux. By comparing these two expressions and by explicitly exhibiting a combinatorial description of level-$2$ affine Weyl group orbits, we establish the existence of a bijection between the Misra-Wilson set of tableaux and a disjoint union of certain sets of multipartitions.

Theorems & Definitions (14)

• Example 2.3.1
• Example 2.3.2
• Theorem 2.3.3
• Example 2.3.4
• Lemma 3.4.1
• Theorem 3.5.1
• Theorem 3.5.2
• Example 3.5.3
• Example 4.2.1
• Theorem 5.1.1