Supercharacter formulas for pattern groups

Authors:
Persi Diaconis and Nathaniel Thiem

Journal:
Trans. Amer. Math. Soc. **361** (2009), 3501-3533

MSC (2000):
Primary 20C99, 05Exx

Published electronically:
March 4, 2009

MathSciNet review:
2491890

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: C. Andre and N. Yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group . In this theory superclasses are certain unions of conjugacy classes, and supercharacters are a set of characters which are constant on superclasses. This paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups.

**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,*The basic character table of the unitriangular group*, J. Algebra**241**(2001), no. 1, 437–471. MR**1839342**, 10.1006/jabr.2001.8734**4.**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**5.**Carlos A. M. André and Ana Margarida Neto,*Super-characters of finite unipotent groups of types 𝐵_{𝑛}, 𝐶_{𝑛} and 𝐷_{𝑛}*, J. Algebra**305**(2006), no. 1, 394–429. MR**2264135**, 10.1016/j.jalgebra.2006.04.030**6.**André, C; Nicolás, A. ``Supercharacters of the adjoint group of a finite radical ring,'' August 2006 preprint.**7.**Antonio Vera-López and J. M. Arregi,*Computing in unitriangular matrices over finite fields*, Linear Algebra Appl.**387**(2004), 193–219. MR**2069276**, 10.1016/j.laa.2004.02.008**8.**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**9.**Roger W. Carter,*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307****10.**Persi Diaconis and I. M. Isaacs,*Supercharacters and superclasses for algebra groups*, Trans. Amer. Math. Soc.**360**(2008), no. 5, 2359–2392. MR**2373317**, 10.1090/S0002-9947-07-04365-6**11.**Persi Diaconis and Laurent Saloff-Coste,*Comparison techniques for random walk on finite groups*, Ann. Probab.**21**(1993), no. 4, 2131–2156. MR**1245303****12.**Isaacs, M. ``Counting characters of upper triangular groups,'' September 2006 preprint.**13.**G. I. Lehrer,*Discrete series and the unipotent subgroup*, Compositio Math.**28**(1974), 9–19. MR**0340438****14.**Richard S. Pierce,*Associative algebras*, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR**674652****15.**Geoffrey R. Robinson,*Counting conjugacy classes of unitriangular groups associated to finite-dimensional algebras*, J. Group Theory**1**(1998), no. 3, 271–274. MR**1633196**, 10.1515/jgth.1998.018**16.**Eugene Spiegel and Christopher J. O’Donnell,*Incidence algebras*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 206, Marcel Dekker, Inc., New York, 1997. MR**1445562****17.**Yan, N.*Representation theory of the finite unipotent linear groups*, Unpublished Ph.D. Thesis, Department of Mathematics, Pennsylvania State Unversity, 2001.**18.**Yan, N. ``Representations of finite unipotent linear groups by the method of clusters,'' 2006 preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20C99,
05Exx

Retrieve articles in all journals with MSC (2000): 20C99, 05Exx

Additional Information

**Persi Diaconis**

Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305-4065

**Nathaniel Thiem**

Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, California 94305-2125

Address at time of publication:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395

DOI:
http://dx.doi.org/10.1090/S0002-9947-09-04521-8

Keywords:
Supercharacters,
superclasses,
finite unipotent group,
algebra group,
posets

Received by editor(s):
October 5, 2006

Received by editor(s) in revised form:
March 1, 2007

Published electronically:
March 4, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

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