 
Summary: arXiv:1009.4134v2[math.CO]23Dec2011
Supercharacters, symmetric functions in noncommuting
variables, and related Hopf algebras
Marcelo Aguiar, Carlos AndrŽe, Carolina Benedetti, Nantel Bergeron,
Zhi Chen, Persi Diaconis, Anders Hendrickson, Samuel Hsiao, I. Martin Isaacs,
Andrea Jedwab, Kenneth Johnson, Gizem Karaali, Aaron Lauve, Tung Le,
Stephen Lewis, Huilan Li, Kay Magaard, Eric Marberg, JeanChristophe Novelli,
Amy Pang, Franco Saliola, Lenny Tevlin, JeanYves Thibon, Nathaniel Thiem,
Vidya Venkateswaran, C. Ryan Vinroot, Ning Yan, Mike Zabrocki
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 fi
nite 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.
1 Introduction
Identifying structures in seemingly disparate fields is a basic task of mathematics. An example,
with parallels to the present work, is the identification of the character theory of the symmetric
group with symmetric function theory. This connection is wonderfully exposited in Macdonald's
