The quotient and forced-lift dictionary for k-Schur and level-restricted structure constants

Definitions and mathematical statement

The workshop asks how $k$-Schur functions relate to $k$-level-restricted Schur functions, specifically whether all level-restricted structure constants occur among the $k$-Littlewood--Richardson coefficients. The source suggests the Goodman--Wenzl root-of-unity Hecke-algebra work as relevant. It does not define its rank, level or quotient conventions.

The supplied aid makes a type-$A$ WZW interpretation. Let $\ell\geq2$, let the Lie rank be $r=\ell-1$, and let $L\geq1$ be the level. The relevant Schur cutoff is \[ K=r+L=\ell+L-1, \] rather than simply $L$. Label the $\operatorname{SU}(\ell)$ level-$L$ fusion basis by partitions $\lambda$ with at most $\ell-1$ rows and $\lambda_1\leq L$. If $[\lambda][\mu]=\sum_\nu N_{\lambda,\mu}^{\nu}[\nu]$, the $N$ are its fusion coefficients. At parameter $1$, define the homogeneous $K$-Schur coefficients by \[ s_\alpha^{(K)}s_\beta^{(K)}=\sum_\gamma c_{\alpha,\beta}^{\gamma,K}s_\gamma^{(K)}. \] For a partition $\lambda$, write $\lambda'$ for its transpose.

The aid attributes to Lapointe--Morse a root-of-unity Hecke quotient in which a $K$-Schur function maps either to its ordinary Schur class or to zero, yielding the desired realization of level-restricted constants. In the transposed fusion convention it reports \[ N_{\lambda,\mu}^{\nu}=c_{\lambda',\mu'}^{(\ell^d,\nu'),K},\qquad d=\frac{|\lambda|+|\mu|-|\nu|}{\ell}, \] when $d$ is a nonnegative integer. Here $(\ell^d,\nu')$ means prepending $d$ full rows of length $\ell$ to $\nu'$. The charge and admissibility conventions for zero coefficients when no such lift exists must be matched to the precise quotient theorem; they are not fully stated in this record.

The aid's homogeneity observation is that the fusion relation $s_{(\ell)}=1$ erases these full rows, while the original homogeneous product forces \[ |\lambda'|+|\mu'|=d\ell+|\nu'|. \] Thus there is at most one nonnegative full-row lift of the indicated type that can contribute. It is the lifted index, rather than necessarily the unchanged index $\nu'$, that recovers the fusion coefficient. For its example $\operatorname{SU}(2)$ at level $2$, the cutoff is $K=3$; the coefficient of the unit in the square of the fundamental class is $1$, corresponding to $c_{(1),(1)}^{(2),3}=1$ with $d=1$, whereas the homogeneous coefficient with output the empty partition is zero.

The relationship is therefore a quotient and index-lifting relationship in this interpretation. The infinite graded $K$-Schur algebra cannot literally be the finite fusion ring with the same grading, since the latter identifies a positive-degree full-row class with the unit. Exact quotient ideals, parameter specialization and the Hecke-to-fusion dictionary remain necessary review data.

Short English statement

The supplied type-$A$ dictionary realizes level-restricted fusion constants as selected $K$-Schur coefficients, with $K$ equal to rank plus level. Homogeneity determines a necessary full-row lift of the output partition, so the relation is a quotient rather than literal equality of the two algebras.

Sources

• Local UnsolvedMath registry, permanent ID $20002905$, source code catalogue entry; canonical aim-special-functions-notes.json, notes[9].
• AIM workshop Generalized Kostka polynomials, Problem (10): https://aimath.org/WWN/kostka/prob2.pdf. No original remarks or separate literature field are supplied.
• Goodman and Wenzl, paper on Iwahori--Hecke algebras of type $A$ at roots of unity, J. Algebra 215 (1999), 694--734, as given in the source. The problem says this paper may be relevant.
• The supplied aid attributes the $K$-Schur quotient theorem to Lapointe--Morse without a full bibliographic reference or quotient presentation.
• Research attempt 1: partial_result, status solved, classification SOLVED-IN-LITERATURE, rigor proved, novelty confidence low. Its forced-lift proposition and literature interpretation are machine-generated aid, not peer reviewed, and require independent verification before citation.