The dual NCSym realization and its internal coproduct

Definitions and mathematical statement

The corrected source asks to describe the graded Hopf isomorphism $SC^{(2)*}\to\Pi^*\cong\mathrm{PiQSym}$ in two presentations and make it compatible with an internal coproduct. The original OCR damages dual stars, indexing, and zero-product conditions. The noncommuting-word presentation uses an alphabet $A=\{a_1,a_2,\ldots\}$ and \[ M_\mu=\sum_{\ker(w)=\mu}w, \] where $\ker(w)$ partitions positions according to equal letters. This realizes $\Pi=\mathrm{NCSym}$. The matrix-variable/cycle-permutation presentation realizes the relevant dual polynomial object; the extract describes monomials of shape $\prod_i x_{i,\sigma(i)}$ with cycle-support partition $\lambda(\sigma)=\mu$, but does not preserve all exact indexing and quotient relations. The two sides must not silently be treated as the same noncommutative algebra.

Use set partitions $\mathcal P_n$, ordered by refinement ($\nu\le\mu$ means finer), and denote their polynomial basis by $U_\mu$. Let $SC^{(2)}$ be the graded superclass-function Hopf algebra for unitriangular groups over ${\mathbb{F}}_2$, with superclass indicators $\kappa_\mu$, and take degreewise graded duals. The aid attributes to Aguiar et al. the outer Hopf isomorphism \[ \Phi(\kappa_\mu^*)=V_\mu,\qquad V_\mu=\sum_{\nu\le\mu}U_\nu. \] The full polynomial presentation and external Hopf operations require the cited theorem, not the corrupted display alone.

The aid makes a specific internal-operation choice: transpose pointwise multiplication of superclass functions. Since $\kappa_\mu\kappa_\nu=\delta_{\mu,\nu}\kappa_\mu$, compatibility uniquely requires \[ \delta(V_\mu)=V_\mu\otimes V_\mu. \] This internal coproduct maps degree $n$ into degree $(n,n)$, unlike the outer Hopf coproduct splitting total degree. With $\mu_{\mathcal P}$ the partition-lattice Möbius function, its reported transported formula is \[ \delta(U_\mu)=\sum_{\nu\le\mu}\mu_{\mathcal P}(\nu,\mu) V_\nu\otimes V_\nu. \] Equivalently, expanding both factors yields \[ \delta(U_\mu)=\sum_{\alpha\vee\beta=\mu}U_\alpha\otimes U_\beta, \] where join is in $\mathcal P_n$. Indeed the coefficient is the finite Möbius sum over common upper bounds below $\mu$. This formula specifies the $U$ expansion without assuming those elements are group-like.

For $\mu=12$, the one-block partition of two elements, the source's explicit test is \[ \delta(U_{12})=U_{1|2}\otimes U_{12} +U_{12}\otimes U_{1|2}+U_{12}\otimes U_{12}. \] Thus $U_{12}$ is not group-like, while $V_{12}$ is. The conclusion is conditional on the pointwise-dual interpretation of "internal"; choosing a different internal operation is a different compatibility question. A complete identification with both source presentations still requires the missing exact polynomial conventions and authoritative reference.

Short English statement

The aid's outer isomorphism sends a dual superclass indicator to a sum of polynomial basis elements. Transposing pointwise multiplication makes those sums group-like and forces a Möbius/join formula for the individual basis elements. The degree-two cross terms rule out the naive group-like rule for the $U$ basis.

Sources

• Local UnsolvedMath registry, permanent ID 20002825, source code catalogue entry; created and updated 14 August 2026. Canonical aim-representation-theory-notes.json, zero-based notes[96], tag problem, item 1.13.
• AIM "Supercharacters and combinatorial Hopf algebras": https://aimath.org/WWN/supercharacters/supercharacters.pdf. Statement status is corrected_verified according to the supplied verification, which reports PDF comparison with the later polynomial realization. The original OCR includes a page footer and damaged quotient conditions; original remarks and literature field are empty.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. It cites Aguiar et al. [AAB] without a full author list, title, publication data, or complete theorem statement.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.