Crystal Bases of Quantum Affine Algebras and Affine Kazhdan-Lusztig Polynomials

TL;DR

This paper introduces a faster algorithm for computing the lower global crystal base of quantum affine sl_n and establishes its coefficients' equivalence with affine Kazhdan-Lusztig polynomials, enabling efficient calculation of decomposition numbers.

Contribution

It presents a faster algorithm for the lower global crystal base and proves its coefficients match affine Kazhdan-Lusztig polynomials, linking two important structures.

Findings

The new algorithm significantly speeds up the computation of crystal bases.

Coefficients of the crystal base coincide with affine Kazhdan-Lusztig polynomials.

Facilitates rapid calculation of decomposition numbers for related algebraic structures.

Abstract

We present a fast version of the algorithm of Lascoux, Leclerc, and Thibon for the lower global crystal base for the Fock representation of quantum affine sl_n. We also show that the coefficients of the lower global crystal base coincide with certain affine Kazhdan-Lusztig polynomials. It is known that the coefficients of the global crystal base are q-analogues of decomposition numbers for Specht modules of the Hecke algebra of type A_n, and that the coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of decomposition numbers for tilting modules for quantum sl_k. Thus our algorithm allows fast computation of these decomposition numbers.