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.
Highlights
Supercharacter theory has infused the representation theory of unipotent groups with the combinatorics of set partitions
Set partitions index the supercharacters of the maximal unipotent upper-triangular subgroup UT of the finite general linear group GL [3, 17], and similar theories exist for the maximal unipotent subgroups of other finite reductive groups [4, 5]
This paper seeks to define a natural family of supercharacter theories for the normal pattern subgroups of UT
Summary
Supercharacter theory has infused the representation theory of unipotent groups with the combinatorics of set partitions. The supercharacters are an orthogonal (but not generally orthonormal) basis for the space of functions that are constant on superclasses This definition has given us new approaches to groups whose representation theories are known to be difficult In addition to teasing out the geometric implications of the underlying polytopes, this paper gives a framework for studying random walks on the integer lattice points of unipotent polytopes, and representation theoretic Hopf structures on them. Both these directions are beyond the scope of this paper
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