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

TL;DR

This paper develops a new family of supercharacter theories for normal pattern groups, revealing combinatorial and geometric structures similar to those of full unipotent groups, thus advancing understanding of their representation theory.

Contribution

It introduces a novel construction of supercharacter theories for normal pattern groups, linking combinatorics and geometry to better understand their representations.

Findings

Supercharacter theories exhibit combinatorial properties akin to set partition combinatorics.

Associated polytopes' lattice points correspond to the theories' geometric structure.

Provides computable character formulas for the constructed theories.

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.