Composition factors of polynomial representation of DAHA and q-decomposition numbers

Abstract

We determine the composition factors of the polynomial representation of DAHA, conjectured by M. Kasatani in [Kasa, Conjecture 6.4.]. He constructed an increasing sequence of subrepresentations in the polynomial representation of DAHA using the “multi-wheel condition'', and conjectured that it is a composition series. On the other hand, DAHA has two degenerate versions called the “degenerate DAHA'' and the “rational DAHA''. The category $\mathcal{O}$ of modules over these three algebras and the category of modules over the $v$-Schur algebra are closely related. By using this relationship, we reduce the determination of composition factors of polynomial representations of DAHA to the determination of the composition factors of the Weyl module $W_v^{(n)}$ 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 $W_v^{(n)}$ by calculating the upper global basis and crystal basis of Fock space of $U_q(\widehat{\mathfrak{sl}}_\ell)$ when $v$ is a primitive $\ell$-th root of unity. This result gives a different way from the determination of decomposition number of $W_v^{(n)}$ by H. Miyachi or B. Ackermann via the modular representation theory of the general linear groups.