Restricting supercharacters of the finite group of unipotent uppertriangular matrices.

Nathaniel Thiem ; Vidya Venkateswaran

The electronic journal of combinatorics, Tome 16 (2009) no. 1

Résumé

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 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).

Détail

DOI : 10.37236/112

Classification : 20G05, 20G40, 05E10
Mots-clés : groups of unipotent upper-triangular matrices, irreducible representations, supercharacters, set-partitions, supercharacter formulas, restrictions