Superinduction for pattern groups

Abstract

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 Diaconis--Isaacs' concept of superinduction in pattern groups. While superinduction shares many desirable properties with usual induction, it no longer takes characters to characters. We begin by finding sufficient conditions guaranteeing that super-induction is in fact induction. It turns out for natural embedding of $U_m$ in $U_n$, super-induction is induction. We conclude with an explicit combinatorial algorithm for computing this induction analogous to the Pieri-formulas for the symmetric group.

References in corpus (3)

• A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic
• Representations of Finite Unipotent Linear Groups by the Method of Clusters
• Irreducible Characters of Finite Algebra Groups

Cited by in corpus (15)

• Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
• A Supercharacter Analogue for Normality
• Exotic characters of unitriangular matrix groups
• Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group
• Iterative character constructions for algebra groups
• Supercharacter theories and Schur rings
• Asymptotic results for Representation Theory
• Restriction and induction for supercharacters of finite groups of triangular type
• Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$
• Construction of supercharacter theories of finite groups
• Restricting supercharacters of the finite group of unipotent uppertriangular matrices
• Towards a supercharacter theory of the parabolic subgroups
• Branching rules in the ring of superclass functions of unipotent upper-triangular matrices
• Shell Tableaux: A set partition analogue of vacillating tableaux
• Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes