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arXiv · 1009.4134

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

Marcelo Aguiar·Carlos Andre·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·Jean-Christophe Novelli·Amy Pang·Franco Saliola·Lenny Tevlin·Jean-Yves Thibon·Nathaniel Thiem·Vidya Venkateswaran·C. Ryan Vinroot·Ning Yan·Mike Zabrocki
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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.

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Marcelo Aguiar, Carlos Andre, 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, Jean-Christophe Novelli, Amy Pang, Franco Saliola, Lenny Tevlin, Jean-Yves Thibon, Nathaniel Thiem, Vidya Venkateswaran, C. Ryan Vinroot, Ning Yan, Mike Zabrocki. 2011-12-23. Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. https://doi.org/10.1016/j.aim.2011.12.024

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