An abelianization supercharacter tower for Sylow subgroups of symmetric groups

Definitions and mathematical statement

The source asks for natural supercharacter theories for Sylow $p$-subgroups of symmetric groups, whether they are nested, and whether restriction splits into supercharacters. Its flattened notation "S pn"/"S pn-1" does not reliably display exponents or the subgroup maps. The aid reads it as the tower on $p^n$ letters with a specified first-block inclusion, distinct from rooted-tree truncation.

Fix a prime $p$ and let $W_0=1$, $W_n=W_{n-1}^{\,p}\rtimes C_p$, with $C_p$ cyclically permuting the factors. This is the standard iterated-wreath Sylow subgroup of $S_{p^n}$. Let $W_{n-1}\hookrightarrow W_n$ be one base-factor inclusion. Its index is $c_n=p^{p^{n-1}}$, and $W_n^{\mathrm{ab}}\cong C_p^n$.

The aid uses the characteristic supercharacter theory whose character blocks are singleton linear characters and one block of all nonlinear irreducibles, omitting the latter if empty. For any finite group $G$ this theory has superclasses $\{1\}$, $G'\setminus\{1\}$ when nonempty, and the nonidentity cosets of $G'$. Define \[ \Lambda_n=\sum_{\lambda\in\operatorname{Irr}(W_n),\,\lambda(1)=1}\lambda, \qquad \Theta_n=\sum_{\chi(1)>1}\chi(1)\chi =\rho_{\mathrm{reg},n}-\Lambda_n. \] Thus $\Theta_1=0$ is a notation convention, not an empty character block counted as a supercharacter.

The reported first-block restriction identity, for $n\ge2$, is \[ \operatorname{Res}_{W_{n-1}}^{W_n}\Theta_n =p^{p^{n-1}}\Theta_{n-1} +\left(p^{p^{n-1}}-p\right)\Lambda_{n-1}. \] Individual linear characters restrict to linear characters, and the sum satisfies $\operatorname{Res}\Lambda_n=p\Lambda_{n-1}$. Hence every supercharacter restricts to a nonnegative integral combination of the lower theory's supercharacters. The degree weights in $\Theta_n$ matter; an unweighted sum of nonlinear characters is a different function.

For the other map, let $W_n\twoheadrightarrow W_{n-1}$ be rooted-tree truncation, deleting the last level. The aid reports that its kernel is not a union of these superclasses for every $n\ge3$. Thus this theory does not descend projectively through those truncations from that level onward. This failure is compatible with the inclusion restriction formula: the two maps and compatibility requirements differ. The construction answers one explicit tower version of the problem; it does not define the intended general notion of natural or nested or classify all possible supercharacter towers.

Short English statement

The aid gives a characteristic theory retaining every linear character and aggregating nonlinear ones. It has an explicit positive restriction rule along first-block inclusions, but fails quotient compatibility under rooted-tree truncation from level three. Those are distinct senses of a compatible tower.

Sources

• Local UnsolvedMath registry, permanent ID 20002820, source code catalogue entry; created and updated 14 August 2026. Canonical aim-representation-theory-notes.json, zero-based notes[91], tag problem, item 1.8.
• AIM "Supercharacters and combinatorial Hopf algebras": https://aimath.org/WWN/supercharacters/supercharacters.pdf. The metadata marks wording exact, although the symmetric-group subscripts/exponents are flattened. Original remarks and literature field are empty.
• Research attempt 1: partial_result, problem status at run open, rigor label proved, novelty confidence low. No complete authoritative references, formal definition of nested, or full general classification is supplied.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.