On the denominators of Young's seminormal basis

TL;DR

This paper investigates the denominators of the base change coefficients between Young's natural basis and the seminormal basis in Specht modules of Iwahori-Hecke algebras, providing new formulas and insights into their structure.

Contribution

It introduces new formulas for these coefficients, especially involving radial lengths, and proves a novel result about summands of restricted Specht modules at roots of unity.

Findings

Derived simple formulas for base change coefficients in key cases.

Established new formulas for general tableaux.

Proved a new result on summands of restricted Specht modules at roots of unity.

Abstract

We study the seminormal basis ${f_t}$ for the Specht modules of the Iwahori-Hecke algebra ${\cal H}_n(q)$ of type $A_{n-1}$. We focus on the base change coefficients between the seminormal basis ${f_t}$ and Young's natural basis ${x_t}$ with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving radial lengths. Even for general tableaux we obtain new formulas. On the way we prove a new result about summands of the restricted Specht module at root of unity.