A prefix-cut obstruction for an NCQSym supercharacter realization

Definitions and mathematical statement

Problem 1.2. Construct a nested supercharacter theory realizing the graded Hopf algebra $\operatorname{NCQSym}$, indexed by ordered set partitions. The source explicitly says "nested" means a projective system of groups, and refers to H. Li's tower-of-algebras work. A complete realization must specify finite groups, supercharacter theories, compatible maps and the product/coproduct operations; a projective system alone does not define those operations.

A supercharacter theory gives a space of superclass functions, with superclass-indicator basis. In the aid's restricted proposed realization, let this basis in degree $n$ be $K_\Phi$, indexed by ordered set partitions $\Phi=(B_1,\ldots,B_k)$ of $[n]$, and require $K_\Phi\mapsto M_\Phi$, the monomial basis of $\operatorname{NCQSym}$. The blocks are nonempty, disjoint and ordered. One monomial convention is to sum words whose equal-letter positions are the blocks and whose distinct letter indices increase in the displayed block order.

Reported exact coproduct constraint. For a sublist of blocks, $\operatorname{std}$ replaces its underlying ordered subset of $[n]$ by consecutive labels, preserving block order. Set the empty list to the degree-zero unit. The required monomial coproduct is \[ \Delta M_\Phi=\sum_{r=0}^k M_{\operatorname{std}(B_1|\cdots|B_r)} \otimes M_{\operatorname{std}(B_{r+1}|\cdots|B_k)}. \] Consequently, under the prescribed basis map, coalgebra compatibility requires exactly this formula for $\Delta K_\Phi$, with coefficients one. Its $(a,n-a)$ component is nonzero precisely when \[ a=|B_1|+\cdots+|B_r| \] for a block prefix. Support alone is only a necessary condition unless coefficients and standardizations also agree; a full Hopf isomorphism additionally needs product/unit/counit compatibility.

For $\Phi=13|2=(\{1,3\},\{2\})$, the internal term is \[ M_{12}\otimes M_1 \] in bidegree $(2,1)$, while bidegree $(1,2)$ is zero. A candidate restriction coproduct symmetric in all unions of blocks, with nonzero all-subset terms, would also yield $K_1\otimes K_{12}$, from selecting the second block first. This forbidden term prevents the prescribed basis map from being a coalgebra isomorphism. Thus a direct block-subset-symmetric refinement of the known $\operatorname{NCSym}$ construction cannot realize this $\operatorname{NCQSym}$ monomial coproduct unchanged.

This is a necessary design obstruction for the specified basis and restriction template, not a nonexistence theorem for every nested supercharacter theory, a different basis change or modified operations. No actual group/algebra categorification is constructed in the supplied aid. It reports that a 2012 BIRS report explicitly left the problem open and that no later solution was found in its search; the full report reference and search date/scope are not supplied. Those are qualified source reports rather than an independent current-status claim.

Short English statement

A basis-preserving NCQSym realization must split ordered blocks only at prefixes. The degree-three partition $13|2$ has a $(2,1)$ coproduct term but no $(1,2)$ term, excluding an unchanged restriction rule symmetric over all block subsets. This restricts one template and does not construct or rule out every nested realization.

Sources

• Local UnsolvedMath registry, permanent ID $20002814$, catalogue entry, aim-representation-theory-notes.json, notes[85], Supercharacters and combinatorial Hopf algebras, May 17--21, 2010, Problem 1.2: https://aimath.org/WWN/supercharacters/supercharacters.pdf.
• Original remark: H. Li, Algebraic structures on Grothendieck groups of a tower of algebras; no publication details are supplied. The remark defines nested as a projective system of groups.
• Supplied research summary invokes an unspecified 2012 BIRS report and gives the prefix-coproduct obstruction. Attempt 1: partial_result, status at run "open", rigor "proved", low novelty confidence; registry status partially_solved. The machine-generated aid is not peer reviewed and requires independent verification before citation; no PDF visit or updated literature search is claimed.