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Published February 20, 2009 | Published
Journal Article Open

Restricting supercharacters of the finite group of unipotent uppertriangular matrices

Abstract

It is well-known that understanding the representation theory of the finite group of unipotent upper-triangular matrices U_n over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where were place partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between Un-1 and Un. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from U_n to U_(n-1) that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions).

Additional Information

© 2009 The Electronic Journal of Combinatorics. Aug 22, 2008; Accepted: Feb 9, 2009; Published: Feb 20, 2009. Part of this work is Venkateswaran's honors thesis at Stanford University. Mathematics Subject Classi�cation: 05E99, 20C33 The authors would like to thank Diaconis and Marberg for many enlightening discussions regarding this work, and anonymous referees for their comments.

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