Quantum Lakshmibai-Seshadri paths and the specialization of Macdonald polynomials at $t=0$ in type $A_{2n}^{(2)}$

Abstract

In this paper, we give a combinatorial realization of the crystal basis of a quantum Weyl module over a quantum affine algebra of type $A_{2n}^{(2)}$, and a representation-theoretic interpretation of the specialization $P_{\lambda}^{A_{2n}^{(2)}} (q,0)$ of the symmetric Macdonald polynomial $P_{\lambda}^{A_{2n}^{(2)}} (q,t)$ at $t=0$, where $\lambda$ is a dominant weight and $P_{\lambda}^{A_{2n}^{(2)}}(q,t)$ denotes the specific specialization of the symmetric Macdonald-Koornwinder polynomial $P_{\lambda}(q,t_1, t_2, t_3, t_4, t_5)$. More precisely, as some results for untwisted affine types, the set of all ($A_{2n}^{(2)}$-type) quantum Lakshmibai-Seshadri paths of shape $\lambda$, which is described in terms of the finite Weyl group $W$, realizes the crystal basis of a quantum Weyl module over a quantum affine algebra of type ${A_{2n}^{(2)}}$ and its graded character is equal to the specialization $P_{\lambda}^{A_{2n}^{(2)}} (q,0)$ of the symmetric Macdonald-Koornwinder polynomial.