Restricting supercharacters of the finite group of unipotent uppertriangular matrices

Details Restricting supercharacters of the finite group of unipotent uppertriangular matrices Restricting supercharacters of the finite group of unipotent uppertriangular matrices

2007-12-07

arxiv.org/abs/0712.1237v1

Mathematics

Representation Theory

Scientific paper

It is well-known that 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 we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. 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).

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