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

Marcelo Aguiar, Carlos André, 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 PangFranco Saliola, Lenny Tevlin, Jean Yves Thibon, Nathaniel Thiem, Vidya Venkateswaran, C. Ryan Vinroot, Ning Yan, Mike Zabrocki

Research output: Contribution to journalArticle

36 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)2310-2337
Number of pages28
JournalAdvances in Mathematics
Volume229
Issue number4
DOIs
StatePublished - Mar 1 2012

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Symmetric Functions
Hopf Algebra
Fourier Analysis
Galois field
Isomorphic
Ring
Coefficient
Class

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Aguiar, M., André, C., Benedetti, C., Bergeron, N., Chen, Z., Diaconis, P., ... Zabrocki, M. (2012). Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. Advances in Mathematics, 229(4), 2310-2337. https://doi.org/10.1016/j.aim.2011.12.024
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. / Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. In: Advances in Mathematics. 2012 ; Vol. 229, No. 4. pp. 2310-2337.
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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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Aguiar, M, André, C, Benedetti, C, Bergeron, N, Chen, Z, Diaconis, P, Hendrickson, A, Hsiao, S, Isaacs, IM, Jedwab, A, Johnson, K, Karaali, G, Lauve, A, Le, T, Lewis, S, Li, H, Magaard, K, Marberg, E, Novelli, JC, Pang, A, Saliola, F, Tevlin, L, Thibon, JY, Thiem, N, Venkateswaran, V, Vinroot, CR, Yan, N & Zabrocki, M 2012, 'Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras', Advances in Mathematics, vol. 229, no. 4, pp. 2310-2337. https://doi.org/10.1016/j.aim.2011.12.024

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

In: Advances in Mathematics, Vol. 229, No. 4, 01.03.2012, p. 2310-2337.

Research output: Contribution to journalArticle

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T1 - Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

AU - Aguiar, Marcelo

AU - André, Carlos

AU - Benedetti, Carolina

AU - Bergeron, Nantel

AU - Chen, Zhi

AU - Diaconis, Persi

AU - Hendrickson, Anders

AU - Hsiao, Samuel

AU - Isaacs, I. Martin

AU - Jedwab, Andrea

AU - Johnson, Kenneth

AU - Karaali, Gizem

AU - Lauve, Aaron

AU - Le, Tung

AU - Lewis, Stephen

AU - Li, Huilan

AU - Magaard, Kay

AU - Marberg, Eric

AU - Novelli, Jean Christophe

AU - Pang, Amy

AU - Saliola, Franco

AU - Tevlin, Lenny

AU - Thibon, Jean Yves

AU - Thiem, Nathaniel

AU - Venkateswaran, Vidya

AU - Vinroot, C. Ryan

AU - Yan, Ning

AU - Zabrocki, Mike

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Aguiar M, André C, Benedetti C, Bergeron N, Chen Z, Diaconis P et al. Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. Advances in Mathematics. 2012 Mar 1;229(4):2310-2337. https://doi.org/10.1016/j.aim.2011.12.024