Algebra-group supercharacter refinements and a class-two formula

Definitions and mathematical statement

The source asks whether the Tung Le--Kay Magaard method of refining supercharacters of $U_n$ has an analogue for arbitrary algebra groups. Its remark points to T. Le's paper Supercharacters and pattern subgroups in the upper triangular groups, without further bibliographic details. The exact refinement method and desired terminal output are not described in the source.

Fix a finite-dimensional nilpotent associative ${\mathbb{F}}_q$-algebra $J$, the group $G=1+J$, a nontrivial additive character $\psi:{\mathbb{F}}_q\to{\mathbb{C}}^*$, and $\lambda\in J^*$. Let $\chi_\lambda$ be the standard Diaconis--Isaacs supercharacter, normalized as the induction of the linear character below. Define \[ \mathfrak l_\lambda=\{x\in J:\lambda(xy)=0\text{ for all }y\in J\}, \] \[ \mathfrak s_\lambda=\{x\in J:\lambda(xy)=0\text{ for all }y\in\mathfrak l_\lambda\}, \qquad L_\lambda=1+\mathfrak l_\lambda,\quad S_\lambda=1+\mathfrak s_\lambda, \] \[ \vartheta_\lambda(1+x)=\psi(\lambda(x))\quad(x\in\mathfrak l_\lambda), \qquad\chi_\lambda=\operatorname{Ind}_{L_\lambda}^G\vartheta_\lambda. \] These definitions fix the left-kernel convention; reversing left/right conventions requires corresponding changes throughout.

The aid reports the 2008 André--Nicolas constituent correspondence and Marberg's iterative development as the natural algebra-group analogue: supercharacters can be refined through canonical algebra subgroups, with induction preserving irreducibility and distinguishing the relevant induced constituents. The general result is a constituent correspondence/refinement mechanism, not a supplied closed classification of all algebra-group irreducibles or a guarantee that no residual subgroup problem remains. The exact general theorem and iterative stopping hypotheses are not reproduced in the record.

Closed class-two formula reported in the aid. Suppose $J^3=0$. Then $\mathfrak l_\lambda$ is an ideal in $\mathfrak s_\lambda$, $S_\lambda/L_\lambda$ is the additive vector space $\mathfrak s_\lambda/\mathfrak l_\lambda$, and the alternating form \[ \beta_\lambda(x+\mathfrak l_\lambda,y+\mathfrak l_\lambda) =\lambda(xy-yx) \] is well defined. Let its radical have ${\mathbb{F}}_q$-dimension $r$ and its rank be $2m$, so $\dim(\mathfrak s_\lambda/\mathfrak l_\lambda)=r+2m$. The form is alternating also in characteristic two; no odd-characteristic assumption is inserted.

The aid reports that the irreducible constituents of $\chi_\lambda$ are exactly $q^r$ distinct characters, each of degree $[G:S_\lambda]q^m$, each occurring with multiplicity $q^m$. Equivalently, \[ \chi_\lambda=q^m\sum_{a=1}^{q^r}\rho_a,\qquad \rho_a\ne\rho_b\ (a\ne b),\qquad \rho_a(1)=[G:S_\lambda]q^m. \] The degree check is \[ q^r q^m[G:S_\lambda]q^m =[G:S_\lambda]q^{r+2m}=[G:L_\lambda]. \] This gives number, common degree and common multiplicity of constituents, not a canonical labeling of all $\rho_a$. It is a special-case explicit decomposition for $J^3=0$, while the general refinement question is addressed by the reported subgroup correspondence. The supplied solved-in-literature label is not an independent theorem verification or novelty claim.

Short English statement

The aid reports an algebra-group refinement correspondence extending the unitriangular idea through canonical subgroups. When the algebra has cube zero, an alternating form gives an explicit decomposition: its radical and rank determine the number, degree and multiplicity of all constituents. This special formula does not amount to a closed classification for arbitrary nilpotent algebras.

Sources

• Local UnsolvedMath registry, permanent ID 20002828, catalogue entry; canonical aim-representation-theory-notes.json, notes[99].
• AIM Supercharacters and combinatorial Hopf algebras, Problem 1.16, https://aimath.org/WWN/supercharacters/supercharacters.pdf. Source names Tung Le and Kay Magaard.
• Original remark: T. Le, Supercharacters and pattern subgroups in the upper triangular groups; no year, journal or URL is supplied.
• Aid attributions: 2008 André--Nicolas constituent correspondence; Marberg's iterative refinement; Diaconis--Isaacs supercharacters. Full references and exact general theorem hypotheses are absent.
• Machine-generated research attempt 1, partial-result classification, reported problem status solved, rigor proved and low novelty confidence; registry classification solved in literature. The class-two formula and refinement reports are not peer reviewed and require independent verification.