Automorphism coarsenings and the reversal-quotient theory of U_n(q)

Definitions and mathematical statement

For a finite group $G$, a supercharacter theory consists of partitions of $G$ into superclasses and of $\operatorname{Irr}(G)$ into character blocks, with equal numbers of blocks, the identity a superclass, and each weighted block sum $\sum_{\chi\in X}\chi(1)\chi$ constant on superclasses. The source asks whether automorphisms of the unitriangular group give useful additional coarsenings, noting that diagonal conjugation removes labels from set partitions. "Nice" and whether to use every automorphism or a chosen subgroup are unspecified.

For any automorphism subgroup $A\le\operatorname{Aut}(G)$, let $\mathcal O_A$ be its orbit supercharacter theory, whose superclasses are $A$-orbits of conjugacy classes, equivalently orbits under $\langle A,\operatorname{Inn}(G)\rangle$. The aid's universal construction is the join $\mathcal S\vee\mathcal O_A$ in the lattice ordered by coarsening: it is the least common coarsening of the given theory and the orbit theory. Here invariance means that each resulting block is setwise fixed, a stronger condition than automorphisms merely permuting blocks. If $A$ permutes the original superclasses and character blocks, their orbit fusion gives this coarsening directly.

The explicit example is $U_n(q)=1+\mathfrak u_n({\mathbb{F}}_q)$, with its standard labeled-set-partition supercharacter theory. Diagonal conjugations independently remove nonzero labels on partition arcs. Let $g^\dagger=Jg^{\mathsf T}J$, with $J$ the antidiagonal permutation matrix, and take the graph automorphism $g\mapsto(g^\dagger)^{-1}$. Together with diagonals it preserves the standard theory and, after labels are removed, sends a partition by reversal $i\mapsto n+1-i$. The aid's coarsening is therefore indexed by ordinary set partitions of $[n]$ modulo reversal. It is field independent at the level of this block indexing, not an assertion that the numerical character values are independent of $q$.

Write $B_n$ for Bell numbers and $F_n$ for the number of reversal-fixed partitions. With $a=\lfloor n/2\rfloor$, $b=n-2a$, and Stirling numbers $\begin{Bmatrix}j\\\ell\end{Bmatrix}$, a weighted Bell--Stirling formula is \[ F_n=\sum_{j=0}^a\binom aj B_{a-j+b} \sum_{\ell=0}^j \begin{Bmatrix}j\\\ell\end{Bmatrix}2^{j-\ell}, \qquad\#\text{blocks}=\frac{B_n+F_n}{2}. \] Use $B_0=1$, $\begin{Bmatrix}0\\0\end{Bmatrix}=1$. The $j$ transposed pairs assigned to exchanged block pairs contribute $2^{j-\ell}$ orientations, while the remaining pairs and possible middle point partition into invariant blocks. Burnside's formula then gives the first counts, for $n=1,\ldots,7$, \[ 1,\ 2,\ 4,\ 11,\ 32,\ 117,\ 468. \] This answers a specific subgroup-fusion construction. It does not classify all automorphisms' action on all supercharacter theories, all possible further coarsenings, or every desirable combinatorial property of the full automorphism join.

Short English statement

The aid gives a universal orbit-join construction and a concrete unitriangular example: diagonal and graph automorphisms produce set partitions modulo reversal. An explicit Bell--Stirling formula counts its blocks. The construction does not classify every full-automorphism coarsening or define the source's word "nice.

Sources

• Local registry, permanent UnsolvedMath ID $20002817$, exact public code catalogue entry; canonical location aim-representation-theory-notes.json, notes[88]. The AIM domain is Representation theory; the registry category is Algebra.
• AIM, Supercharacters and combinatorial Hopf algebras, Problem 1.5; supplied reference https://aimath.org/WWN/supercharacters/supercharacters.pdf. The original note says torus conjugation lets one ignore partition labels. The extracted page-12 editor header names Carolina Benedetti. No PDF was accessed here.
• John A. Gibbs, Automorphisms of certain unipotent groups, as suggested by the source; no year, publication data, or theorem number is supplied.
• Supplied research attempt 1: partial_result, run and registry status partially_solved, rigor label proved, low novelty confidence; classification PARTIAL-PROGRESS, created and updated 14 August 2026. The aid is machine generated, not peer reviewed, and requires independent verification before citation.