Narrow Results

We prove that the quantized Carter–Lusztig basis for a finite-dimensional irreducible U_q(𝔤𝔩_n(ℂ))-module V(λ) is related to the global crystal basis for V(λ) by an upper triangular invertible matrix. We express the global crystal basis in terms of the q-Schur algebra and provide an algorithm for obtaining global crystal basis vectors for V(λ) using the q-Schur algebra.

Consider the affine Lie algebra $\widehat{s\ell }(n)$ with null root $\delta$, weight lattice $P$ and set of dominant weights $P^{+}$. Let $V(k{\mathrm{\Lambda }}_0),$$k\in {\mathbb{Z}}_{\ge 1}$ denote the integrable highest weight $\widehat{s\ell }(n)$-module with level $k\ge 1$ highest weight $k{\mathrm{\Lambda }}_0$. Let $\mathrm{\text{wt}}(V)$ denote the set of weights of $V(k{\mathrm{\Lambda }}_0)$. A weight $\mu \in \mathrm{\text{wt}}(V)$ is a maximal weight if $\mu +\delta \notin \mathrm{\text{wt}}(V)$. Let ${\mathrm{\text{max}}}^{+}(k{\mathrm{\Lambda }}_0)=\mathrm{\text{max}}(k{\mathrm{\Lambda }}_0)\cap P^{+}$ denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set ${\mathrm{\text{max}}}^{+}(k{\mathrm{\Lambda }}_0)$ is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of $\widehat{sl}(n)$-modules, Algebr. Represent. Theory17 (2014) 1303–1321]. In papers [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb.22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the $\widehat{s\ell }(n)$-modules $V(k{\mathrm{\Lambda }}_0)$, Algebr. Represent. Theory25 (2022) 477–490], the multiplicities of certain subsets of ${\mathrm{\text{max}}}^{+}(k{\mathrm{\Lambda }}_0)$ were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the $\widehat{s\ell }(n)$-module $V(k{\mathrm{\Lambda }}_0)$ are given generalizing the results in [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb.22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the $\widehat{s\ell }(n)$-modules $V(k{\mathrm{\Lambda }}_0)$, Algebr. Represent. Theory25 (2022) 477–490].