Quantum $\mathfrak{gl}$-weight system and its average values

TL;DR

This paper proves a conjecture about the average value of the universal -weight system on permutations, using quantum algebra techniques to connect it to Schur functions and Bernoulli polynomials.

Contribution

It introduces a quantum analogue of the -weight system, providing a deformation that links average values to Schur functions and q-analogues of Bernoulli polynomials.

Findings

Proved the conjecture on average values of the -weight system.

Connected the quantum weight system to Schur functions and Bernoulli polynomials.

Established a deformation framework using Hecke algebras.

Abstract

We present a proof of a recent conjecture due to M. Kazarian, E. Krasilnikov, S. Lando, and M. Shapiro, which describes the average value of the universal $\mathfrak{gl}$-weight system on permutations. The proof uses a quantum analogue of the $\mathfrak{gl}$-weight system on Hecke algebras of type $A$, which leads to a one-parameter deformation of the average value of the universal ${\mathfrak{gl}}$-weight system. We show that the average value of the quantum weight system is a linear combination of one-part Schur functions, with coefficients being $q$-analogues of Bernoulli polynomials.