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

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 isomor...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2012-03, Vol.229 (4), p.2310-2337
Main Authors: Aguiar, Marcelo, André, Carlos, Benedetti, Carolina, Bergeron, Nantel, Chen, Zhi, Diaconis, Persi, Hendrickson, Anders, Hsiao, Samuel, Isaacs, I. Martin, Jedwab, Andrea, Johnson, Kenneth, Karaali, Gizem, Lauve, Aaron, Le, Tung, Lewis, Stephen, Li, Huilan, Magaard, Kay, Marberg, Eric, Novelli, Jean-Christophe, Pang, Amy, Saliola, Franco, Tevlin, Lenny, Thibon, Jean-Yves, Thiem, Nathaniel, Venkateswaran, Vidya, Vinroot, C. Ryan, Yan, Ning, Zabrocki, Mike
Format: Article
Language:English
Subjects:
Combinatorial Hopf algebra
Combinatorics
Finite fields
Mathematics
Representation Theory
Supercharacters
Symmetric functions in noncommuting variables
Unipotent uppertriangular matrices
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2011.12.024