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 Pang; Franco Saliola; Lenny Tevlin; Jean Yves Thibon; Nathaniel Thiem; Vidya Venkateswaran; C. Ryan Vinroot; Ning Yan; Mike Zabrocki

doi:10.1016/j.aim.2011.12.024

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

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

54 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 language	English
Pages (from-to)	2310-2337
Number of pages	28
Journal	Advances in Mathematics
Volume	229
Issue number	4
DOIs	https://doi.org/10.1016/j.aim.2011.12.024
Publication status	Published - 1 Mar 2012

User-Defined Keywords

Supercharacters
Unipotent uppertriangular matrices
Symmetric functions in noncommuting variables
Finite fields
Combinatorial Hopf algebra

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10.1016/j.aim.2011.12.024

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Aguiar, M., André, C., Benedetti, C., Bergeron, N., Chen, Z., Diaconis, P., Hendrickson, A., Hsiao, S., Isaacs, I. M., Jedwab, A., Johnson, K., Karaali, G., Lauve, A., Le, T., Lewis, S., Li, H., Magaard, K., Marberg, E., Novelli, J. C., ... 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

@article{bf44bb6120fb49d1b782c135994da91a,

title = "Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras",

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.",

keywords = "Supercharacters, Unipotent uppertriangular matrices, Symmetric functions in noncommuting variables, Finite fields, Combinatorial Hopf algebra",

author = "Marcelo Aguiar and Carlos Andr{\'e} and Carolina Benedetti and Nantel Bergeron and Zhi Chen and Persi Diaconis and Anders Hendrickson and Samuel Hsiao and Isaacs, {I. Martin} and Andrea Jedwab and Kenneth Johnson and Gizem Karaali and Aaron Lauve and Tung Le and Stephen Lewis and Huilan Li and Kay Magaard and Eric Marberg and Novelli, {Jean Christophe} and Amy Pang and Franco Saliola and Lenny Tevlin and Thibon, {Jean Yves} and Nathaniel Thiem and Vidya Venkateswaran and Vinroot, {C. Ryan} and Ning Yan and Mike Zabrocki",

note = "Funding information: This paper developed during a focused research week at the American Institute of Mathematics in May 2010. The main results presented here were proved as a group during that meeting. Publisher copyright: {\textcopyright} 2012 Elsevier Inc. All rights reserved.",

year = "2012",

month = mar,

day = "1",

doi = "10.1016/j.aim.2011.12.024",

language = "English",

volume = "229",

pages = "2310--2337",

journal = "Advances in Mathematics",

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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

TY - JOUR

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

N1 - Funding information: This paper developed during a focused research week at the American Institute of Mathematics in May 2010. The main results presented here were proved as a group during that meeting. Publisher copyright: © 2012 Elsevier Inc. All rights reserved.

PY - 2012/3/1

Y1 - 2012/3/1

N2 - 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.

AB - 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.

KW - Supercharacters

KW - Unipotent uppertriangular matrices

KW - Symmetric functions in noncommuting variables

KW - Finite fields

KW - Combinatorial Hopf algebra

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-84863069286&partnerID=MN8TOARS

U2 - 10.1016/j.aim.2011.12.024

DO - 10.1016/j.aim.2011.12.024

M3 - Journal article

SN - 0001-8708

VL - 229

SP - 2310

EP - 2337

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 4

ER -

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

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