Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes

Abstract

Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of the combinatorial properties of the set partition combinatorics of the full uni-triangular groups, including combinatorial indexing sets, dimensions, and computable character formulas. Associated with these supercharacter theories is also a family of polytopes whose integer lattice points give the theories geometric underpinnings.

References in corpus (6)

• Hopf monoids from class functions on unitriangular matrices
• Representations of Finite Unipotent Linear Groups by the Method of Clusters
• A Supercharacter Analogue for Normality
• Supercharacters of unipotent groups defined by involutions
• A supercharacter table decomposition via power-sum symmetric functions
• Closed expressions for averages of set partition statistics