Borel supercharacters and a split torus-degree Hopf structure

Definitions and mathematical statement

For a prime power $q$, let $U_n({\mathbb{F}}_q)$ be the upper unitriangular subgroup and $B_n({\mathbb{F}}_q)$ the upper triangular invertible subgroup of $\operatorname{GL}_n({\mathbb{F}}_q)$. The workshop asks to carry the $U_n$ supercharacter constructions over to $B_n$: is there a similar "(module to a) Hopf algebra"? The supplied source verification treats this parenthetical as an editorial ambiguity already present in the source. Accordingly, retain both targets: a Hopf algebra of Borel superclass functions, and a module attached to an appropriate Hopf algebra. The phrase "all of the $U_n$ work" does not identify a finite list of constructions to extend.

The aid reports that Panov's 2018 work supplies a natural supercharacter theory of $B_n({\mathbb{F}}_q)$ for which \[ H=\bigoplus_{n\geq0}\operatorname{SCF}(B_n({\mathbb{F}}_q)) \] is a graded Hopf algebra isomorphic to partially symmetric functions in noncommuting variables. Here $\operatorname{SCF}$ means the complex span of superclass indicators for that particular theory, and the degree-zero term is ${\mathbb{C}}$. Neither the full supercharacter theory nor formulas for the Hopf operations are included in this record; arbitrary induction and restriction operations are not implicitly asserted to give this structure.

For that reported Hopf algebra, the further assertions in the aid are as follows. Its rigged-partition basis has indices describable by a subset $R\subseteq[n]$ of nonidentity torus decorations, a choice of label in ${\mathbb{F}}_q^\times\setminus\{1\}$ at each point of $R$, and a set partition of $[n]\setminus R$. Let $H_{n,r}$ be the span of basis elements with $|R|=r$. The claimed bigrading satisfies \[ H=\bigoplus_{n\geq0}\bigoplus_{r=0}^n H_{n,r},\qquad H_{n,r}H_{m,s}\subseteq H_{n+m,r+s},\qquad \Delta H_{n,r}\subseteq\bigoplus_{a+b=n,\,u+v=r}H_{a,u}\otimes H_{b,v}. \] Writing $\operatorname{Bell}(k)$ for the number of set partitions of a $k$-element set, including $\operatorname{Bell}(0)=1$, the reported dimensions are \[ \dim_{\mathbb{C}} H_{n,r}=\binom nr(q-2)^r\operatorname{Bell}(n-r). \] For $q=2$ the only nonzero torus degree is $r=0$; the factor with exponent zero is interpreted as $1$.

Let $A=\operatorname{NCSym}$ be the Hopf algebra of symmetric functions in noncommuting variables, identified in the aid with the diagonal-averaged unitriangular Hopf algebra. The aid asserts a Hopf inclusion $i:A\longrightarrow H$ onto $\bigoplus_n H_{n,0}$ and a Hopf projection $p:H\longrightarrow A$ satisfying \[ p\circ i=\operatorname{id}_A,\qquad p(H_{n,r})=0\quad(r>0). \] In its function model, $p$ evaluates all nonidentity torus letters at zero. Thus $H$ has the left $A$-module structure $a\cdot h=i(a)h$ and left $A$-comodule structure $(p\otimes\operatorname{id})\Delta$. The corresponding asserted Radford biproduct is \[ H\simeq K\mathbin{\#} A,\qquad K=\{h\in H:(\operatorname{id}\otimes p)\Delta(h)=h\otimes1\}, \] where $K$ carries the associated braided Hopf structure. These are supplied literature and research-aid assertions, requiring verification of the precise basis, operations and identifications.

Short English statement

Extend unitriangular supercharacter constructions to finite Borel groups. The source's ambiguous wording asks for either a Hopf algebra or an associated module; the supplied aid reports a Borel Hopf algebra and a split projection to $\operatorname{NCSym}$, with a refined dimension formula.

Sources

• Local UnsolvedMath registry, permanent ID $20002822$, source code catalogue entry; canonical aim-representation-theory-notes.json, notes[93].
• AIM Problem Lists, Representation theory, workshop Supercharacters and combinatorial Hopf algebras, Problem 1.10, supplied PDF source: https://aimath.org/WWN/supercharacters/supercharacters.pdf. No remarks or original literature references are supplied. The source-verification field reports that "(module to a)" is original wording; the registry statement status is reconstructed_unverified.
• Panov's 2018 work is cited by name and year in the supplied research summary for the Borel Hopf algebra and partially symmetric function identification; no title or complete bibliographic citation is supplied.
• Research attempt 1 is labeled partial_result, problem status solved, classification SOLVED-IN-LITERATURE, rigor proved, and novelty confidence low. Its bigrading, projection and biproduct claims are machine-generated research aid, not peer reviewed, and require independent verification before citation.