Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-19T00:56:14.655Z Has data issue: false hasContentIssue false
  • English
  • Français

Supercharacters and pattern subgroups in the upper triangular groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  21 November 2012

Tung Le
Tung Le*
Affiliation:
Faculty of Mathematics and Computer Science, Vietnam National University, Ho Chi Minh City, Vietnam (lttung96@yahoo.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Un(q) denote the upper triangular group of degree n over the finite field with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form Uk(q) ∩ wUk(q), where w is a monomial matrix in GLk(q) for some k < n.

Keywords

root systemirreducible charactertriangular group

MSC classification

Secondary: 20C33: Representations of finite groups of Lie type 20C15: Ordinary representations and characters
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 177 - 186
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.André, C. A. M., Basic characters of the unitriangular group (for arbitrary primes), Proc. Am. Math. Soc. 130 (2002), 1943–1954.CrossRefGoogle Scholar
2.André, C. A. M. and Neto, A. M., Supercharacters of finite unipotent groups of types Bn, Cn and Dn, J. Alg. 305 (2006) 394–429.CrossRefGoogle Scholar
3.Arregi, J. M., and Vera-López, A., Conjugacy classes in Sylow p-subgroups of GL(n, q), J. Alg. 152(1) (1992), 1–19.Google Scholar
4.Carter, R. W., Simple groups of Lie type, Pure and Applied Mathematics (Wiley, 1972).Google Scholar
5.Diaconis, P. and Isaacs, I. M., Supercharacters and superclasses for algebra groups, Trans. Am. Math. Soc. 360(5) (2008), 2359–2392.CrossRefGoogle Scholar
6.Diaconis, P. and Thiem, N., Supercharacter formulas for pattern groups, Trans. Am. Math. Soc. 361 (2009), 3501–3533.CrossRefGoogle Scholar
7.Evseev, A., Reduction for characters of finite algebra groups, J. Alg. 325 (2011), 321–351.CrossRefGoogle Scholar
8.Higman, G., Enumerating p-groups, I, Inequalities, Proc. Lond. Math. Soc. 10 (1960), 24–30.CrossRefGoogle Scholar
9.Huppert, B., Character theory of finite groups (de Gruyter, Berlin, 1998).CrossRefGoogle Scholar
10.Isaacs, I. M., Characters of groups associated with finite algebras, J. Alg. 177 (1995) 708–730.CrossRefGoogle Scholar
11.Isaacs, I. M., Counting characters of upper triangular groups, J. Alg. 315 (2007), 698–719.CrossRefGoogle Scholar
12.Le, T., Counting irreducible representations of large degree of the upper triangular groups, J. Alg. 324(8) (2010), 1803–1817.CrossRefGoogle Scholar
13.Lehrer, G. I., Discrete series and the unipotent subgroup, Compositio Math. 28 (1974), 9–19.Google Scholar
14.Robinson, G., Counting conjugacy classes of unitriangular groups associated to finite dimensional algebras, J. Group Theory 1 (1998), 271–274.CrossRefGoogle Scholar
15.Thompson, J., k(Un(Fq)), Preprint (available at www.math.ufl.edu/fac/thompson.html).Google Scholar
16.Thompson, J., (Us(F), Ur(F)) double consets in U(F), Papers I–III, Preprints (available at www.math.ufl.edu/fac/thompson.html).Google Scholar