Green formula apps

This is a set of pages that allow you 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.

Please let me know if you find any bugs or if you have any ideas for features you would like to have!

• 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 $\mathrm{GL}_4(\mathbb{F}_q)$.