Quantum invariants for decomposition problems in type A rings of representations

TL;DR

This paper introduces a combinatorial rule for decomposing certain p-adic $GL_n$ representations, simplifying calculations and confirming a conjecture, with applications to quantum algebra and KLR modules.

Contribution

It provides a new combinatorial rule for decomposition that avoids Kazhdan-Lusztig polynomial computations and confirms Lapid's conjecture.

Findings

Established a combinatorial decomposition rule for ladder representations.

Eliminated the need for Kazhdan-Lusztig polynomial calculations.

Extended results to quantum affine algebra modules.

Abstract

We prove a combinatorial rule for a complete decomposition, in terms of Langlands parameters, for representations of p-adic $GL_n$ that appear as parabolic induction from a large family (ladder representations). Our rule obviates the need for computation of Kazhdan-Lusztig polynomials in these cases, and settles a conjecture posed by Lapid. These results are transferrable into various type A frameworks, such as the decomposition of convolution products of homogeneous KLR-algebra modules, or tensor products of snake modules over quantum affine algebras. The method of proof applies a quantization of the problem into a question on Lusztig's dual canonical basis and its embedding into a quantum shuffle algebra, while computing numeric invariants which are new to the p-adic setting.