Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group
Eric Marberg
DOI: 10.1007/s10801-011-0293-5
Abstract
Let U n denote the group of n\times n unipotent upper-triangular matrices over a fixed finite field \mathbb F q \mathbb{F}_{q}, and let U P U_{\mathcal{P}} denote the pattern subgroup of U n corresponding to the poset P \mathcal{P}. This work examines the superclasses and supercharacters, as defined by Diaconis and Isaacs, of the family of normal pattern subgroups of U n . After classifying all such subgroups, we describe an indexing set for their superclasses and supercharacters given by set partitions with some auxiliary data. We go on to establish a canonical bijection between the supercharacters of U P U_{\mathcal{P}} and certain \mathbb F q \mathbb {F}_{q}-labeled subposets of P \mathcal{P}. This bijection generalizes the correspondence identified by André and Yan between the supercharacters of U n and the \mathbb F q \mathbb{F}_{q}-labeled set partitions of {1,2,\cdots , n}. At present, few explicit descriptions appear in the literature of the superclasses and supercharacters of infinite families of algebra groups other than { U n : n\in \Bbb N}. This work significantly expands the known set of examples in this regard.
Pages: 61–92
Keywords: keywords unitriangular group; pattern groups; algebra groups; supercharacter theories; supercharacters; superclasses; labeled posets; labeled set partitions
Full Text: PDF
References
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2. André, C.A.M.: The basic character table of the unitriangular group. J. Algebra 241, 437-471 (2001)
3. André, C.A.M.: Basic characters of the unitriangular group (for arbitrary primes). Proc. Am. Math. Soc. 130, 1934-1954 (2002)
4. Arias-Castro, E., Diaconis, P., Stanley, R.: A super-class walk on upper-triangular matrices. J. Algebra 278, 739-765 (2004)
5. Diaconis, P., Isaacs, M.: Supercharacters and superclasses for algebra groups. Trans. Am. Math. Soc. 360, 2359-2392 (2008)
6. Diaconis, P., Thiem, N.: Supercharacter formulas for pattern groups. Trans. Am. Math. Soc. 361, 3501-3533 (2009)
7. Hendrickson, A.: Construction of supercharacter theories of finite groups. Preprint. (2009)
8. Lewis, S., Thiem, N.: Nonzero coefficients in restrictions and tensor products of supercharacters of Un(q). Adv. Math. 227, 40-72 (2011)
9. Marberg, E.: A supercharacter analogue for normality. J. Algebra 332, 334-365 (2011)
10. Marberg, E., Thiem, N.: Superinduction for pattern groups. J. Algebra 321, 3681-3703 (2009)
11. Shapiro, L.: Upper triangular rings, ideals, and Catalan numbers. Am. Math. Mon. 82, 634-637 (1975)
12. Thiem, N.: Branching rules in the ring of superclass functions of unipotent upper-triangular matrices.
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