Hilbert-series obstructions to supercharacter Hopf quotients of Sym

Definitions and mathematical statement

The workshop asks whether the four infinite families of supercharacter theories of the symmetric groups $S_n$ can be realized as Hopf quotients of $\operatorname{Sym}$, and whether multiplication and comultiplication can be defined on one of these families to give a Hopf algebra. For a specified family of superclass partitions $\mathcal K_n$, its graded vector space would be \[ H=\bigoplus_{n\geq0}\operatorname{SCF}(S_n;\mathcal K_n),\qquad \operatorname{SCF}(S_n;\mathcal K_n)=\{f:S_n\to{\mathbb{C}}:f\text{ is constant on each superclass}\}, \] with the usual degree-zero convention $H_0={\mathbb{C}}$. The record does not define the four partitions $\mathcal K_n$ or specify the desired operations, so their reconstruction must remain explicit qualification rather than part of the verified original statement.

The supplied aid uses the ordinary character family, the maximal supercharacter family, and coarse and fine $A_n$ star-product families reconstructed from the context of Problems 1.17--1.18. Its characteristic-zero obstruction is this: for a field $k$ of characteristic zero, \[ \operatorname{Sym}_k=k[p_1,p_2,\ldots],\qquad \deg p_r=r,\qquad \Delta p_r=p_r\otimes1+1\otimes p_r. \] Every connected graded Hopf quotient, and every graded Hopf subalgebra, of this Hopf algebra has Hilbert series \[ \operatorname{Hilb}(H;t)=\prod_{r\in A}(1-t^r)^{-1} \] for some subset $A\subseteq\{1,2,\ldots\}$. Thus, writing the uniquely determined formal Euler expansion of a candidate dimension series as \[ \sum_{n\geq0}\dim H_n\,t^n=\prod_{r\geq1}(1-t^r)^{-a_r}, \] a necessary condition for such a quotient or subalgebra is $a_r\in\{0,1\}$ for every $r$. There can be at most one retained primitive generator in each positive degree.

Under its stated reconstruction, the aid reports that the ordinary family gives $\operatorname{Sym}$; that both the maximal family and the coarse $A_n$ star-product family violate this Hilbert-series condition in degree $4$; and that the fine $A_n$-orbit star-product family has $a_8=2$, with its first primitive-multiplicity obstruction in degree $8$. It also reports that the usual induction product \[ f\star g=\operatorname{Ind}_{S_m\times S_n}^{S_{m+n}}(f\boxtimes g) \] fails to preserve the maximal-family space already in total degree $3$. These are qualified assertions about the reconstructed families, whose precise definitions and small-degree conventions are missing from this record. Obstructions to a graded quotient or subalgebra of $\operatorname{Sym}$, or to that inherited product, do not settle the separate existence question for other possible Hopf operations.

Short English statement

Ask whether the four symmetric-group supercharacter families give Hopf quotients of $\operatorname{Sym}$ or admit suitable Hopf operations. The supplied characteristic-zero test permits zero or one primitive generator in each degree and reportedly excludes the three proper reconstructed families as such quotients or subalgebras.

Sources

• Local UnsolvedMath registry, permanent ID $20002830$, source code catalogue entry; canonical aim-representation-theory-notes.json, notes[101].
• AIM Problem Lists, Representation theory, workshop Supercharacters and combinatorial Hopf algebras, Problem 1.18: https://aimath.org/WWN/supercharacters/supercharacters.pdf. No original remarks or literature references are supplied. The registry labels the statement exact, but its verification field ends with "The canonical JSON record reads:" without another quotation.
• Supplied research attempt 1: partial_result, problem status at run open, current registry status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. Its reconstruction refers to Problems 1.17--1.18. No authoritative external citation for the classification or obstructions is supplied; the machine-generated aid is not peer reviewed and requires independent verification before citation.