Elementary characters generate Andre's supercharacter theory

Definitions and mathematical statement

The source asks whether André's supercharacter theory is the coarsest retaining the single-box elementary irreducibles. Its reference to irreducible "superclasses" is imprecise, since irreducibility is a character property, and "the irreducibles" could mean this distinguished family or all irreducible characters. The aid uses the former reading and retains every field-labeled elementary character separately.

For a finite group $G$, a complex supercharacter theory consists of partitions $\mathcal X$ of $\operatorname{Irr}(G)$ and $\mathcal K$ of $G$ into superclasses, with the standard identity, equal-cardinality, and constancy conditions for \[ \sigma_X=\sum_{\chi\in X}\chi(1)\chi. \] Its superclass-function space $\operatorname{SCF}$ is the unital pointwise algebra of functions constant on $\mathcal K$, spanned by these supercharacters. To retain an irreducible $\chi$ means that $\{\chi\}$ is a character block, not merely that it occurs as a constituent somewhere. Coarsest means that every competing theory retaining the specified family refines its superclass and character partitions.

The aid's general criterion is: if a supercharacter theory $\mathcal S_0$ retains a family $E$ of singleton irreducibles and \[ \operatorname{SCF}(\mathcal S_0)={\mathbb{C}}[E] \] as a unital pointwise algebra, then it is the unique coarsest theory retaining $E$. Its superclasses are exactly the joint value fibers \[ g\sim h\quad\Longleftrightarrow\quad \chi(g)=\chi(h)\text{ for every }\chi\in E. \] Indeed the comparison concerns the function algebra generated by fixed labeled characters, not unlabeled or phase-modified value vectors.

For $G=U_n(q)$, the upper-unitriangular group over ${\mathbb{F}}_q$, take $n\ge1$ and all elementary characters $\chi_{ij,a}$ with $1\le i<j\le n$ and $a\in{\mathbb{F}}_q^\times$. The aid reports that these are singleton irreducibles in the André--Yan theory, and every basic supercharacter associated to a labeled rook placement is the pointwise product of its single-box elementary characters. Since the basic supercharacters span its superclass-function space, the criterion gives the asserted unique coarsest property for every prime power $q$. This statement uses all labels and all allowed single positions, not only first-superdiagonal linear characters.

If the final source phrase instead requires every irreducible character of $G$ to be a singleton block, the theory is simply ordinary character theory with conjugacy classes as superclasses. That is a different reading. The aid's reported solution addresses the specified elementary-family interpretation; the exact original intent and cited product theorem still require review.

Short English statement

Under the explicit reading that all labeled elementary irreducibles must remain singleton blocks, the aid reports that their pointwise products generate André--Yan's entire superclass algebra, making it the unique coarsest such theory. Requiring all irreducibles instead would give ordinary character theory.

Sources

• Local UnsolvedMath registry, permanent ID 20002815, source code catalogue entry; created and updated 14 August 2026. Canonical aim-representation-theory-notes.json, zero-based notes[86], tag problem, item 1.3.
• AIM "Supercharacters and combinatorial Hopf algebras": https://aimath.org/WWN/supercharacters/supercharacters.pdf. Wording is marked exact; original remarks and literature field are empty. The source names André's theory without a complete bibliographic citation.
• Research attempt 1: full_solution, problem status at run solved, rigor label proved, novelty confidence low. Registry labels are solved and SOLVED-IN-LITERATURE. No complete authoritative reference or product-formula theorem statement for the André--Yan theory is supplied.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.