Supercharacters, symmetric functions in noncommuting variables (extended abstract)

• Published in: Discrete Mathematics & Theoretical Computer Science
• 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
• Other Authors: Department of Mathematics and Statistics Texas Tech, Texas Tech University Lubbock (TTU), Université de Lisbonne, Department of Mathematics and Statistics Toronto, York University Toronto, Department of Statistics Stanford, Stanford University, Concordia College MN, Bard College, University of Wisconsin-Madison, University of Southern California (USC), Pennsylvania State University (Penn State), Penn State System, Pomona College, Loyola University Chicago, University of Aberdeen, University of Washington Seattle, Computer Science Department Drexel, Drexel University, University of Birmingham Birmingham, MIT Laboratory for Computer Science, Laboratoire d'Informatique Gaspard-Monge (LIGM), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout (BEZOUT), Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de combinatoire et d'informatique mathématique Montréal (LaCIM), Centre de Recherches Mathématiques Montréal (CRM), Université de Montréal (UdeM)-Université de Montréal (UdeM)-Université du Québec à Montréal = University of Québec in Montréal (UQAM), New York University New York (NYU), NYU System (NYU), Department of Mathematics, University of Colorado, University of Colorado Boulder, California Institute of Technology (CALTECH), College of William and Mary Williamsburg (WM), Unknown, Bousquet-Mélou, Mireille and Wachs, Michelle and Hultman, Axel
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2011
• Subjects: supercharacters; set partitions; symmetric functions in non-commuting variables; Hopf algebras; [INFO]Computer Science [cs]; [MATH]Mathematics [math]; Iceland
• Online Access: https://inria.hal.science/hal-01336771; https://inria.hal.science/hal-01336771/document; https://inria.hal.science/hal-01336771/file/dmAO0102.pdf; https://doi.org/10.46298/dmtcs.2967

International audience 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. Nous montrons que deux structures en apparence bien différentes peuvent être identifiées: les super-caractères, qui sont un outil commode pour faire de l'analyse de Fourier sur le groupe des matrices unipotentes triangulaires supérieures à coefficients dans un corps fini, et l'anneau des fonctions symétriques en variables non-commutatives. Ces deux structures sont des algèbres de Hopf isomorphes. Cette identification permet de traduire dans une structure les dévelopements conçus pour l'autre, et suggère de nombreux exemples dans le domaine nouveau des algèbres de Hopf combinatoires.