Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

Carolina Benedetti
Nantel Bergeron
Zhi Chen
C. Ryan Vinroot, William & Mary

Abstract

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras. (C) 2012 Elsevier Inc. All rights reserved.