Finite general linear groups

Applications for representation theory of $\mathrm{GL}_n(\mathbb{F}_q)$

Green polynomials

The app Green polynomials allows one to compute Green polynomials for $\mathrm{GL}_n\left( \mathbb{F}_q \right)$ . It uses Theorem 2.6 from the article The Green polynomials via vertex operators by Naihuan Jing and Ning Liu.

Characters tables of $\mathrm{GL}_n(\mathbb{F}_q)$

The following set of pages allows one to explore the character tables of $\mathrm{GL}_n(\mathbb{F}_q)$ . It implements the formula established by Green in The characters of the finite general linear groups.

Representation list - allows one to inspect all the different types of representations of $\mathrm{GL}_n(\mathbb{F}_q)$ for a given $n$ . Try $n = 2$ to see the classical classification of irreducible representations of $\mathrm{GL}_2(\mathbb{F}_q)$ !
Conjugacy class list - allows one to inspect all the different types of conjugacy classes of $\mathrm{GL}_n(\mathbb{F}_q)$ for a given $n$ . Try $n = 2$ to see the classical classification of conjugacy classes of $\mathrm{GL}_2(\mathbb{F}_q)$ !

These two pages link to the following pages.

Character formula - allows one to view Green's character formula for a given representation. For example, the input (1, [1], \alpha), (1, [1], \beta) corresponds to an irreducible generic principal series representation of $\mathrm{GL}_2(\mathbb{F}_q)$ , while the input (4, [1], \theta) corresponds to an irreducible cuspidal representation of $\mathrm{GL}_4(\mathbb{F}_q)$ .
Conjugacy class formula - allows one to view Green's character formula for a given conjugacy class. For example, the input (1, [1], x), (1, [1], y) corresponds to a diagonal matrix in $\mathrm{GL}_2(\mathbb{F}_q)$ , while the input (4, [1], \xi) corresponds to a regular elliptic element in $\mathrm{GL}_4(\mathbb{F}_q)$ .

Generealized Gelfand-Graev multiplicities

The Character formula page links to Gelfand-Graev multiplicities. This pages displays the multiplicities of a given representation in every possible generalized Gelfand--Graev representation. It is based on Sections 3.1.1-3.1.2 and Proposition 5.7 of the paper The combinatorics of $\mathrm{GL}_n$ generalized Gelfand–Graev characters by Scott Andrews and Nathaniel Thiem. See for example the multiplicities of the following two Speh representations for $\mathrm{GL}_6(\mathbb{F}_q)$ : $\Delta(\tau_3, 2)$ , $\Delta(\tau_2, 3)$ .