The Green polynomials via vertex operators

TL;DR

This paper introduces new formulas and iterative methods for Green polynomials using vertex operator techniques, enhancing understanding of their combinatorial structure and connections to representation theory.

Contribution

It provides a general combinatorial formula, compact formulas for specific cases, a Murnaghan-Nakayama type formula, and an iterative approach for related algebraic structures.

Findings

Derived an iterative formula for Green polynomials.

Established compact formulas for cases with partitions of length ≤ 3.

Developed a Murnaghan-Nakayama type formula for Green polynomials.

Abstract

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.