Unipotent Hecke algebras of GL_n(F_q)

TL;DR

This paper introduces a family of Hecke algebras associated with unipotent subgroups of GL_n(F_q), providing combinatorial bases, commutative subalgebras, and a generalization of the RSK correspondence relevant to their representation theory.

Contribution

It characterizes the structure of unipotent Hecke algebras of GL_n(F_q), including bases, commutative subalgebras, and a generalized RSK correspondence, advancing understanding of their representation theory.

Findings

Combinatorial basis for H_μ established

Identification of a large commutative subalgebra

Generalization of the RSK correspondence for these algebras

Abstract

This paper describes a family of Hecke algebras H_\mu=End_G(Ind_U^G(\psi_\mu)), where U is the subgroup of unipotent upper-triangular matrices of G=GL_n(F_q) and \psi_\mu is a linear character of U. The main results combinatorially index a basis of H_\mu, provide a large commutative subalgebra of H_\mu, and after describing the combinatorics associated with the representation theory of H_\mu, generalize the RSK correspondence that is typically found in the representation theory of the symmetric group.