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. PreliminariesThis paper is primarily concerned with unipotent subgroups of the finite general linear groups. It is standard to index the rows and columns of the corresponding matrices by the set t1, 2, . . . , N u in the usual total order, but recent work [2] has suggested that it is better to instead allow an arbitrary set of size N to index the rows and columns and fix a total order N on that set. However, the paper may be easily read with N as the total order 1 ă 2 㨨¨ă N .This section reviews the relevant unipotent groups, a combinatorial interpretation of normality, and some of the standard supercharacter theories for these groups. The last section gives a quick refresher of q-binomial coefficients.

Supercharacters for parabolic contractions of finite groups of A,B,C,D Lie types

“…Also, we classify supercharacters and superclasses for the unipotent radical U a (see Theorems 3.3 and 3.6). For the parabolic radicals of GL(n), the similar classification is verified in [16].…”

“…In our case, G = U a and a classification of (U a × U a )-orbits in u a and (u a ) * is considered an extremely difficult problem (see [10,14,16]). Therefore, we consider the more "coarse" supercharacter theory (see Theorem 3.6) in which superclasses are attached to the (GL a (n) × GL a (n))-orbits in u a , and supercharacters a up to a constant multiple equal to sums of functions ε µ(x) , where µ runs through a (GL a (n) × GL a (n))-orbit in (u a ) * .…”

Two supercharacter theories for the parabolic subgroups in orthogonal and symplectic groups

“…Suppose that r km = r ′ km and d k = d ′ k for any ℓ k > m −ℓ, then as in Item 2.1 they coincide under the action of block-diagonal permutation matrix. ✷In the case of type A, the similar statement follows from the proof of[18, Theorem 4.2].We attach to each basic subset D and map φ :D → F q the element in u * of the form λ D,φ = γ∈D φ(γ)E * γ ,where {E * γ } is a dual basis for the basis {E γ } in u. The following statement is analogical to Proposition 3.3.…”

The combinatorics of normal subgroups in the unipotent upper triangular group

“…Uniformly indexing the conjugacy classes or irreducible characters of UT n (F q ) (for every n and q) is an impossibly difficult problem [13]. However, developments in unipotent combinatorics [1,2,5,29] suggest that the difficulty of the conjugacy problem belies nicer combinatorial structures present in the more computable, conjugacy-adjacent properties of UT n (F q ). Absent a good understanding of the fundamental structures in the representation theory of UT n (F q ), two questions arise: what methods can access the important properties of UT n (F q ), and is there a nice, combinatorial description for the output of these methods?…”