Kazhdan-Lusztig Basis and Optimization

TL;DR

This paper introduces an optimization-based approach to understanding canonical bases of Hecke algebras, connecting algebraic structures with quadratic optimization and invariant cones.

Contribution

It formulates a novel optimization framework to recover and analyze Kazhdan--Lusztig and other positive bases, extending to representations of rak{sl}_n.

Findings

Maximal invariant cones contain the Kazhdan--Lusztig basis for certain shapes.

Minimizing the Gram matrix trace recovers Young's seminormal basis.

Optimization detects deviations from Kazhdan--Lusztig basis in higher ranks.

Abstract

We describe a conjectural approach to obtaining canonical bases of the Hecke algebra at $q=1$ via continuous quadratic optimization. We focus on Specht modules $S^\lambda$ and proper cones inside $S^\lambda$ that are invariant under the action of $1+s$ for all simple reflections $s\in S$. We show that there are unique minimal and maximal cones invariant under all $1+s$. For hook shapes, two-column shapes, and partitions of the form $(n-2,2)$, we prove that the Kazhdan--Lusztig basis spans this maximal cone. More generally, we define an optimization problem over bases that are unitriangular with respect to the polytabloid basis, subject to the constraint that the operators $1+s$ act non-negatively. We prove that the feasible region forms a compact semialgebraic set, and interpret it in terms of a hierarchy of invariant cones under all $1+s$. We demonstrate that minimizing the trace of the Gram matrix uniquely recovers Young's seminormal basis. Furthermore, we verify computationally that maximization uniquely recovers the Kazhdan--Lusztig basis for all partitions of $n\leq 7$. In higher ranks, the optimization detects deviations from the Kazhdan--Lusztig basis and may favour other natural positive bases, such as the Springer basis or $p$-canonical bases. Finally, we extend this framework to irreducible representations of $\mathfrak{sl}_n$. We observe that the Gelfand--Tsetlin basis corresponds to the unique minimizer, and we conjecture that the canonical basis corresponds to the maximum in small ranks.