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

الأصل · EN

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 slₙ. 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 Aₙ, and that the coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of decomposition numbers for tilting modules for quantum slₖ. Thus our algorithm allows fast computation of these decomposition numbers.