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

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.