Supercharacters and superclasses for algebra groups

Authors:
Persi Diaconis and I. M. Isaacs

Journal:
Trans. Amer. Math. Soc. **360** (2008), 2359-2392

MSC (2000):
Primary 20C15, 20D15

Published electronically:
November 20, 2007

MathSciNet review:
2373317

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups. These groups generalize the unimodular upper triangular groups over a finite field, and the supercharacter theory we develop extends results of Carlos André and Ning Yan that were originally proved in the upper triangular case. This theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table. We discuss connections with the Kirillov orbit method and with Gelfand pairs, and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class, respectively. We also show that products of supercharacters are positive integer combinations of supercharacters.

**1.**Carlos A. M. André,*Basic characters of the unitriangular group*, J. Algebra**175**(1995), no. 1, 287–319. MR**1338979**, 10.1006/jabr.1995.1187**2.**Carlos A. M. André,*Irreducible characters of finite algebra groups*, Matrices and group representations (Coimbra, 1998) Textos Mat. Sér. B, vol. 19, Univ. Coimbra, Coimbra, 1999, pp. 65–80. MR**1773571****3.**Carlos A. M. André,*Basic characters of the unitriangular group (for arbitrary primes)*, Proc. Amer. Math. Soc.**130**(2002), no. 7, 1943–1954 (electronic). MR**1896026**, 10.1090/S0002-9939-02-06287-1**4.**Ery Arias-Castro, Persi Diaconis, and Richard Stanley,*A super-class walk on upper-triangular matrices*, J. Algebra**278**(2004), no. 2, 739–765. MR**2071663**, 10.1016/j.jalgebra.2004.04.005**5.**I. Martin Isaacs,*Character theory of finite groups*, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR**1280461****6.**I. M. Isaacs,*Characters of groups associated with finite algebras*, J. Algebra**177**(1995), no. 3, 708–730. MR**1358482**, 10.1006/jabr.1995.1325**7.**I. M. Isaacs, Algebra groups, unpublished notes, 1997.**8.**I. M. Isaacs and Dikran Karagueuzian,*Conjugacy in groups of upper triangular matrices*, J. Algebra**202**(1998), no. 2, 704–711. MR**1617655**, 10.1006/jabr.1997.7311**9.**I. M. Isaacs and Dikran Karagueuzian,*Erratum: “Conjugacy in groups of upper triangular matrices” [J. Algebra 202 (1998), no. 2, 704–711; MR1617655 (99b:20011)]*, J. Algebra**208**(1998), no. 2, 722. MR**1655475**, 10.1006/jabr.1998.7430**10.**Andrei Jaikin-Zapirain,*A counterexample to the fake degree conjecture*, Chebyshevskiĭ Sb.**5**(2004), no. 1(9), 188–192. MR**2098978****11.**A. A. Kirillov,*Variations on the triangular theme*, Lie groups and Lie algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI, 1995, pp. 43–73. MR**1364453****12.**I. G. Macdonald,*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144****13.**Josu Sangroniz,*Characters of algebra groups and unitriangular groups*, Finite groups 2003, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 335–349. MR**2125084****14.**Ning Yan, Representation Theory of the finite unipotent linear groups, unpublished manuscript, 2001.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20C15,
20D15

Retrieve articles in all journals with MSC (2000): 20C15, 20D15

Additional Information

**Persi Diaconis**

Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall Bldg. 380, Stanford, California 94305

Email:
diaconis@math.stanford.edu

**I. M. Isaacs**

Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison, Wisconsin 53706

Email:
isaacs@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04365-6

Received by editor(s):
December 30, 2005

Published electronically:
November 20, 2007

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.