Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

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 $ U_n(\mathbb{F}_q)$. 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.

References [Enhancements On Off] (What's this?)

  • 1. André, C. ``Basic characters of the unitriangular group,'' Journal of algebra 175 (1995), 287-319. MR 1338979 (96h:20081a)
  • 2. André, C. ``Irreducible characters of finite algebra groups,'' Matrices and group representations Coimbra, 1998, Textos Mat. Sér B 19 (1999), 65-80. MR 1773571 (2001g:20009)
  • 3. André, C. ``The basic character table of the unitriangular group,'' Journal of Algebra 241 (2001), 437-471. MR 1839342 (2002e:20082)
  • 4. André, C. ``Basic characters of the unitriangular group (for arbitrary primes),'' Proceedings of the American Mathematical Society 130 (2002), 1934-1954. MR 1896026 (2003g:20075)
  • 5. André, C; Neto, A. ``Super-characters of finite unipotent groups of types $ B_n$, $ C_n$ and $ D_n$,'' J. Algebra 305 (2006), 394-429. MR 2264135 (2007j:20013)
  • 6. André, C; Nicolás, A. ``Supercharacters of the adjoint group of a finite radical ring,'' August 2006 preprint.
  • 7. Arregi, J; Vera-Lopez, A. ``Computing in unitriangular matrices over finite fields.'' Linear algebra applications 387 (2004), 193-219. MR 2069276 (2005c:20082)
  • 8. Arias-Castro, E; Diaconis, P; Stanley, R. ``A super-class walk on upper-triangular matrices,'' Journal of Algebra 278 (2004), 739-765. MR 2071663 (2005f:60101)
  • 9. Carter, R. Finite groups of Lie type: Conjugacy classes and complex characters. John Wiley and Sons, 1985. MR 794307 (87d:20060)
  • 10. Diaconis, P; Isaacs, M. ``Supercharacters and superclasses for algebra groups,'' Transactions of the American Mathematical Society, 360 (2008), 2359-2392. MR 2373317
  • 11. Diaconis, P; Saloff-Coste, L. ``Comparison techniques for random walk on finite groups,'' Annals of Probability 21 (1993), 2131-2156. MR 1245303 (95a:60009)
  • 12. Isaacs, M. ``Counting characters of upper triangular groups,'' September 2006 preprint.
  • 13. Lehrer, G. ``Discrete series and the unipotent subgroup,'' Compositio Mathematica 28 (1974), 9-19. MR 0340438 (49:5193)
  • 14. Pierce, R. Associative Algebras, Graduate Texts in Mathematics 88. Studies in the History of Modern Science 9. Springer-Verlag, New York-Berlin, 1982. MR 674652 (84c:16001)
  • 15. Robinson, G. ``Counting conjugacy classes of unitriangular groups associated to finite-dimensional algebras,'' Journal of Group Theory 1 (1998), 271-274. MR 1633196 (99h:14025)
  • 16. Spiegel, E; O'Donnell, C. Incidence algebras, Monographs and textbooks in pure mathematics 206, Marcel Dekker, Inc., New York: 1997. MR 1445562 (98g:06001)
  • 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.

Similar Articles

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

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.

American Mathematical Society