Composition Factors of Polynomial Representation of DAHA and Crystallized Decomposition Numbers

Abstract

We determine the composition factors of the polynomial representation of DAHA, conjectured by M. Kasatani. We reduce the determination of composition factors of polynomial representations of DAHA to the determination of the composition factors of the Weyl module $W^{(1^n)}$ for the $v$-Schur algebra. By using the LLT-Ariki type theorem of $v$-Schur algebra proved by Varagnolo-Vasserot, we determine the composition factors of $W^{(1^n)}$ by calculating the upper global basis and crystal basis of Fock space of $U_q(\hat{\mf{sl}}_\ell)$.