An elementary-generator characterization of the André–Yan theory

Definitions and mathematical statement

Problem 1.4 asks to characterize the André--Yan supercharacter theory "along the lines of the previous problem". The preceding local registry record, permanent ID 20002815, asks whether André's theory is the coarsest supercharacter theory containing the single-box elementary irreducibles individually. This supplies a natural minimality interpretation, rather than an unspecified characterization. The neighboring record is context and is not altered or completed here.

Fix a prime power $q$, $n\ge1$, $G=U_n(q)$, and the algebra $J$ of strictly upper-triangular matrices over ${\mathbb{F}}_q$, so $G=1+J$. Fix a nontrivial additive character $\psi:{\mathbb{F}}_q\to{\mathbb{C}}^*$. In the standard algebra-group formulation the André--Yan superclasses are \[ 1+G XG\quad(X\in J). \] For $\lambda\in J^*$, the associated supercharacter can be normalized as \[ \chi_\lambda(g)=\frac{|G\lambda|}{|G\lambda G|} \sum_{\nu\in G\lambda G}\psi(\nu(g-1)). \] The dual left/right actions are those induced by multiplication on $J$. The single-box elementary characters are $\xi_{ij}(a)=\chi_{\lambda_{ij,a}}$, where $i<j$, $a\in{\mathbb{F}}_q^*$, and $\lambda_{ij,a}(X)=aX_{ij}$. The aid uses the known irreducibility of these elementary characters. For $n=1$ the set is empty and the group is trivial.

A supercharacter theory consists of a partition of $G$ into superclasses and a compatible partition of $\operatorname{Irr}(G)$, with singleton identity superclass and the usual equal-size and character-constancy conditions. Retaining an elementary irreducible individually means that its singleton is a block of the irreducible-character partition, not merely that it occurs somewhere in a reducible supercharacter. Coarser theories have coarser superclass partitions.

Characterization reported in the supplied aid. Let $\operatorname{SC}_{\mathrm{AY}}(G)$ be the complex algebra of all functions constant on André--Yan superclasses, with pointwise multiplication. For every $n$ and prime power $q$, the aid reports \[ \operatorname{SC}_{\mathrm{AY}}(G) ={\mathbb{C}}[\xi_{ij}(a):i<j,\ a\in{\mathbb{F}}_q^*], \] where the right side is the unital pointwise function algebra. Equivalently, for $g,h\in G$, \[ g,h\text{ lie in the same Andr\'e--Yan superclass} \quad\Longleftrightarrow\quad \xi_{ij}(a)(g)=\xi_{ij}(a)(h) \text{ for every }i<j,\ a\in{\mathbb{F}}_q^*. \] Consequently the André--Yan theory is the unique coarsest supercharacter theory retaining all these elementary irreducibles as singleton character blocks. This is a characterization relative to a distinguished prescribed family, not a classification of all supercharacter theories of $U_n(q)$.

The aid attributes the underlying input to André's arbitrary-characteristic factorization and unique-support results: basic supercharacters factor as products of elementary single-box characters, and irreducible supports form the required partition. The finite-function-algebra interpretation then identifies superclasses with simultaneous value fibers. Full authoritative references and normalization details for those inputs are not supplied. The assertion is a reported supplied result, while the registry remains partially solved and the attempt's problem-status field says open; neither metadata label is independently resolved here.

Short English statement

Using the preceding problem's minimality question, the aid characterizes the André–Yan theory by its single-box elementary characters. They generate its superclass-function algebra, so their simultaneous value fibers are exactly the superclasses. Equivalently it is the coarsest theory keeping those irreducibles individually, rather than a classification of every supercharacter theory.

Sources

• Local UnsolvedMath registry, permanent ID 20002816, catalogue entry; canonical aim-representation-theory-notes.json, notes[87]. Neighboring local record 20002815, AIM-REPRESENTATION_THEORY-0087, supplies Problem 1.3's single-box/coarsest-theory context.
• AIM Supercharacters and combinatorial Hopf algebras, Problem 1.4, https://aimath.org/WWN/supercharacters/supercharacters.pdf. The aid reports agreement with the original PDF; this draft does not independently check it. No individual proposer is supplied.
• André and Yan are named by the source theory; the aid invokes André's arbitrary-characteristic factorization and unique-support theorem without full bibliographic entries.
• Machine-generated research attempt 1, partial result, attempt status open, reported rigor proved and low novelty confidence; registry status partially solved. Its generator/fiber/minimality assertion is not peer reviewed and requires independent verification before citation.