# Polynomiality of Grothendieck groups for finite general linear groups, Deligne-Lusztig characters, and injective unstable modules

## TL;DR

This paper proves that the Grothendieck groups of finitely generated projective modules over finite general linear groups form a polynomial algebra, with explicit generators linked to Deligne-Lusztig characters, revealing algebraic structure and connections.

## Contribution

It establishes the polynomiality of the Grothendieck groups for finite general linear groups and explicitly describes the generators and their relation to Deligne-Lusztig characters.

## Key findings

- The algebra formed by Grothendieck groups is polynomial.
- Explicit generators are identified and related to Deligne-Lusztig characters.
- The structure facilitates understanding of module categories over finite groups.

## Abstract

Let K 0 (Fp GLn(Fp)-proj) denote the Grothendieck group of finitely generated pro-jective Fp GLn(Fp)-modules. We show that the algebra C $\otimes$ n$\ge$0 K 0 (Fp GLn(Fp)-proj) with multiplication given by induction functors, is a polynomial algebra. We explicit generators and their relation with Deligne-Lusztig characters.