A Plethysm formula on the characteristic map of induced linear characters from $U_n(\mathbb F_q)$ to $GL_n(\mathbb F_q)$

TL;DR

This paper derives a plethysm formula for the characteristic map of induced linear characters from unipotent upper-triangular matrices to general linear groups over finite fields, linking it to Hall-Littlewood symmetric functions.

Contribution

It introduces a plethysm formula involving twisted Hall-Littlewood functions for induced characters from $U_n(F_q)$ to $GL_n(F_q)$ and provides a recurrence relation for computation.

Findings

The characteristic map relates to a multiple of twisted Hall-Littlewood functions.

A recurrence relation simplifies calculations of the plethysm formula.

The formula advances understanding of induced characters in finite general linear groups.

Abstract

This paper gives a plethysm formula on the characteristic map of the induced linear characters from the unipotent upper-triangular matrices $U_n(\mathbb F_q)$ to $GL_n(\mathbb F_q)$, the general linear group over finite field $\mathbb F_q$. The result turns out to be a multiple of a twisted version of the Hall-Littlewood symmetric functions $\tilde{P}_n(Y,q)$. A recurrence relation is also given which makes it easy to carry out the computation.