The Symbolic and cancellation-free formulae for Schur elements

TL;DR

This paper introduces new symbolic and cancellation-free formulas for Schur elements in degenerate cyclotomic Hecke algebras, enabling easier analysis of their properties and applications.

Contribution

It provides the first explicit symbolic and cancellation-free formulas for Schur elements in this algebra class, enhancing understanding and computational efficiency.

Findings

Schur elements are symmetric under natural group action.

Schur elements are polynomials with integer coefficients.

A new proof of semi-simplicity criterion for these algebras.

Abstract

In this paper we give a symbolical formula and a cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some direct applications, we show that the Schur elements are symmetric with respect to the natural symmetric group action and are integral coefficients polynomials and we give a different proof of Ariki-Mathas-Rui's criterion on the semi-simplicity of degenerate cyclotomic Hecke algebras.