Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths

Let $\lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(\lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $\lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $\lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $\lambda$, $\mu$, we have an isomorphism $\Theta : \mathrm{QLS}(\lambda + \mu) \rightarrow \mathrm{QLS}(\lambda) \otimes \mathrm{QLS}(\mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $\Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.