Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases

TL;DR

This paper connects Kazhdan-Lusztig bases with Pfaffians and discriminants, providing new algebraic expressions and specializations that relate to symmetric group representations and $q$-discriminants.

Contribution

It introduces a novel expression of Pfaffians using Kazhdan-Lusztig bases and links these to $q$-discriminants through specific parameter specializations.

Findings

Expressed Pfaffians with symmetric group bases.

Derived $q$-discriminant from specialized parameters.

Established algebraic relations between bases and discriminants.

Abstract

We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries $(a_i-a_j) h_{i+j}$. In the case where the parameters $a_i$ are specialized to successive powers of $q$, and the $h_i$ are complete functions, we obtain the $q$-discriminant.