A rigging-refined stable-level fusion polynomial

Definitions and mathematical statement

The source considers the level-$k$ fusion product of rectangular type A factors \[ B=B^{r_\ell,s_\ell}\otimes_k\cdots\otimes_k B^{r_1,s_1} \] and asks for a two-variable $(q,t)$ generalization of its grading. It reports type A combinatorial/fermionic fusion formulas through rigged configurations (Schilling et al.) and conjectural other-type formulas (Hatayama et al.). The target multiplicity, energy normalization and meaning of a second grading are not fixed; a positive generating polynomial alone is weaker than a module bigrading.

The aid uses type $A_n^{(1)}$ rectangular crystals, $1\le r_j\le n$, $s_j\ge1$, and fixes a dominant classical weight $\lambda$. Let $L_i^{(a)}$ count factors $B^{a,i}$. A configuration consists of partitions $\nu^{(a)}$, with $m_i^{(a)}$ parts of size $i$, satisfying \[ \sum_{a,i}i m_i^{(a)}\alpha_a =\sum_{a,i}iL_i^{(a)}\omega_a-\lambda. \] For the type A Cartan matrix $C$, define vacancy numbers and configuration cocharge by \[ p_i^{(a)}=\sum_j\min(i,j)L_j^{(a)} -\sum_b C_{ab}\sum_j\min(i,j)m_j^{(b)}, \] \[ \operatorname{cc}(\nu)=\tfrac12\sum_{a,b,i,j} C_{ab}\min(i,j)m_i^{(a)}m_j^{(b)}. \] For an admissible configuration, each rigging $J_i^{(a)}$ is a partition fitting an $m_i^{(a)}\times p_i^{(a)}$ rectangle. Put $R(\nu,J)=\sum_{a,i}|J_i^{(a)}|$.

Fix the aid's energy-compatible rigged-configuration bijection $\Psi$, obtained from the chosen standard bijection by the appropriate corigging complement, and normalize the path energy by \[ D(b)=\operatorname{cc}(\nu)+R(\nu,J),\qquad (\nu,J)=\Psi(b). \] This convention must be matched to the one-variable fusion polynomial; a change of additive energy normalization changes the prefactor.

At stable level $k\ge\sum_j s_j$, the aid reports the positive refinement \[ F_{B,\lambda}(q,t)=\sum_{b\text{ highest weight }\lambda} q^{D(b)}t^{R(\Psi(b))} =\sum_{\nu\text{ admissible}}q^{\operatorname{cc}(\nu)} \prod_{a,i} \begin{bmatrix}p_i^{(a)}+m_i^{(a)}\\m_i^{(a)}\end{bmatrix}_{qt}. \] The Gaussian binomial at $u$ counts partitions in the indicated rectangle by $u^{|J|}$. Thus the expression has nonnegative coefficients. In the reported normalization, $t=1$ gives the usual one-dimensional $X=M$ polynomial, and $q=t=1$ gives the corresponding fusion coefficient. The second degree is invariant under changing factor order through the combinatorial $R$-matrix and the compatible rigged-configuration identification, not through arbitrary unrelated bijections.

For a fixed configuration put $A_\nu=\sum_{a,i}m_i^{(a)}p_i^{(a)}$. Complementing each rectangle sends $R$ to $A_\nu-R$. If $P_\nu(u)$ is the product of Gaussian binomials, the exact fiber reciprocity is \[ u^{A_\nu}P_\nu(u^{-1})=P_\nu(u). \] Equivalently the fiber $q^{\operatorname{cc}(\nu)}P_\nu(qt)$ is reciprocal with factor $q^{2\operatorname{cc}(\nu)+A_\nu}t^{A_\nu}$ under $(q,t)\mapsto(q^{-1},t^{-1})$. The factor depends on $\nu$; no global single-degree reciprocity is asserted.

For arbitrary fusion level, let $\mathrm{RC}_k$ be the established level-restricted set in the standard bijection convention, and let $\theta$ complement riggings. The aid's energy-side set is $\theta(\mathrm{RC}_k)$, not automatically $\mathrm{RC}_k$ itself. Summing $q^{\operatorname{cc}(\nu)+R}t^R$ over this image gives its finite-level positive candidate. Level restrictions can couple riggings, so the stable Gaussian product and complement reciprocity do not automatically persist. The summary does not supply the exact finite-level inequalities, a closed product formula, a representation-theoretic bigrading, or identification with a canonical Macdonald $(q,t)$ theory. Those remain additional parts of the original request.

Short English statement

The aid proposes total energy-side rigging size as a second statistic. At stable fusion level this gives a positive Gaussian-binomial formula with the correct one-variable and ungraded specializations. Finite-level restrictions require the complemented restricted set and need not factor. The construction is a combinatorial refinement, not a supplied module bigrading or canonical Macdonald deformation.

Sources

• Local UnsolvedMath registry, permanent ID 20002907, catalogue entry; canonical aim-special-functions-notes.json, notes[11], reconstructed-unverified statement status.
• AIM Generalized Kostka polynomials, Problem 12, https://aimath.org/WWN/kostka/prob2.pdf. No individual proposer is supplied.
• Source attributions: Schilling et al. for type A rigged-configuration interpretations and fermionic fusion formulas; Hatayama et al. for conjectural other-type formulas. No full bibliographic entries are supplied, and the source's conjectural wording is a historical report rather than an independently updated status.
• Machine-generated research attempt 1, partial-result/partially-solved labels, reported rigor proved and low novelty confidence. Its second statistic and stable-level formula are not peer reviewed and require independent verification.