Supercharacters and superclasses for algebra groups

Diaconis, Persi; Isaacs, I.

doi:10.1090/S0002-9947-07-04365-6

Supercharacters and superclasses for algebra groups
HTML articles powered by AMS MathViewer

by Persi Diaconis and I. M. Isaacs PDF

Trans. Amer. Math. Soc. 360 (2008), 2359-2392 Request permission

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.

References

Carlos A. M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995), no. 1, 287–319. MR 1338979, DOI 10.1006/jabr.1995.1187
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
Carlos A. M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (2002), no. 7, 1943–1954. MR 1896026, DOI 10.1090/S0002-9939-02-06287-1
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, DOI 10.1016/j.jalgebra.2004.04.005
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
I. M. Isaacs, Characters of groups associated with finite algebras, J. Algebra 177 (1995), no. 3, 708–730. MR 1358482, DOI 10.1006/jabr.1995.1325
I. M. Isaacs, Algebra groups, unpublished notes, 1997.
I. M. Isaacs and Dikran Karagueuzian, Conjugacy in groups of upper triangular matrices, J. Algebra 202 (1998), no. 2, 704–711. MR 1617655, DOI 10.1006/jabr.1997.7311
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, DOI 10.1006/jabr.1998.7430
Andrei Jaikin-Zapirain, A counterexample to the fake degree conjecture, Chebyshevskiĭ Sb. 5 (2004), no. 1(9), 188–192. MR 2098978
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, DOI 10.1090/trans2/169/05
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
Josu Sangroniz, Characters of algebra groups and unitriangular groups, Finite groups 2003, Walter de Gruyter, Berlin, 2004, pp. 335–349. MR 2125084
Ning Yan, Representation Theory of the finite unipotent linear groups, unpublished manuscript, 2001.