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

Division of Engineering and Science (Abington)

Research output: Contribution to journal › Article › peer-review

41 Scopus citations

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 (US)
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
State	Published - Mar 1 2012

All Science Journal Classification (ASJC) codes

Mathematics(all)

Access to Document

10.1016/j.aim.2011.12.024

Cite this

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

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.

@article{303b94cbf657473f90399d2ff603a54a,

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.",

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: diaconis@math.stanford.edu (P. Diaconis), ahendric@cord.edu (A. Hendrickson), hsiao@bard.edu (S. Hsiao), isaacs@math.wisc.edu (I.M. Isaacs), jedwab@usc.edu (A. Jedwab), kwj1@psu.edu (K. Johnson), gizem.karaali@pomona.edu (G. Karaali), lauve@math.luc.edu (A. Lauve), lttung96@yahoo.com (T. Le), stedalew@u.washington.edu (S. Lewis), huilan.li@gmail.com (H. Li), k.magaard@bham.ac.uk (K. Magaard), emarberg@math.mit.edu (E. Marberg), novelli@univ-mlv.fr (J.-C. Novelli), amypang@stanford.edu (A. Pang), saliola@gmail.com (F. Saliola), ltevlin@nyu.edu (L. Tevlin), jyt@univ-mlv.fr (J.-Y. Thibon), thiemn@colorado.edu (N. Thiem), vidyav@caltech.edu (V. Venkateswaran), vinroot@math.wm.edu (C.R. Vinroot), ning.now@gmail.com (N. Yan), zabrocki@mathstat.yorku.ca (M. Zabrocki). 1 Supported by NSF DMS-1001935. 2 Supported by CRC and Beca Mazda. 3 Supported by CRC and NSERC. 4 Supported by NSF DMS-0804324. 5 Supported by NSF DMS 07-01291. 6 Supported by NSA H98230-10-1-0362. 7 Supported by NSF DMS-0854893. 8 Supported by NSF DMS-0652641. 9 Supported by NDSEG Fellowship. 10 Supported by CRC. 11 Supported by NSF DMS-0854849. 12 Supported by NSERC.",

year = "2012",

month = mar,

day = "1",

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

language = "English (US)",

volume = "229",

pages = "2310--2337",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press Inc.",

number = "4",

}

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 journal › Article › peer-review

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: diaconis@math.stanford.edu (P. Diaconis), ahendric@cord.edu (A. Hendrickson), hsiao@bard.edu (S. Hsiao), isaacs@math.wisc.edu (I.M. Isaacs), jedwab@usc.edu (A. Jedwab), kwj1@psu.edu (K. Johnson), gizem.karaali@pomona.edu (G. Karaali), lauve@math.luc.edu (A. Lauve), lttung96@yahoo.com (T. Le), stedalew@u.washington.edu (S. Lewis), huilan.li@gmail.com (H. Li), k.magaard@bham.ac.uk (K. Magaard), emarberg@math.mit.edu (E. Marberg), novelli@univ-mlv.fr (J.-C. Novelli), amypang@stanford.edu (A. Pang), saliola@gmail.com (F. Saliola), ltevlin@nyu.edu (L. Tevlin), jyt@univ-mlv.fr (J.-Y. Thibon), thiemn@colorado.edu (N. Thiem), vidyav@caltech.edu (V. Venkateswaran), vinroot@math.wm.edu (C.R. Vinroot), ning.now@gmail.com (N. Yan), zabrocki@mathstat.yorku.ca (M. Zabrocki). 1 Supported by NSF DMS-1001935. 2 Supported by CRC and Beca Mazda. 3 Supported by CRC and NSERC. 4 Supported by NSF DMS-0804324. 5 Supported by NSF DMS 07-01291. 6 Supported by NSA H98230-10-1-0362. 7 Supported by NSF DMS-0854893. 8 Supported by NSF DMS-0652641. 9 Supported by NDSEG Fellowship. 10 Supported by CRC. 11 Supported by NSF DMS-0854849. 12 Supported by NSERC.

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.

UR - http://www.scopus.com/inward/record.url?scp=84863069286&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863069286&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2011.12.024

DO - 10.1016/j.aim.2011.12.024

M3 - Article

AN - SCOPUS:84863069286

VL - 229

SP - 2310

EP - 2337

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this