arXiv subjects

Fumihiko Nomoto

Publications and source records attributed to Fumihiko Nomoto.

5 recordsLinked to original sources

arXiv2022

Symmetric and Nonsymmetric Macdonald Polynomials via a Path Model with a Pseudo-crystal Structure

In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating in standard monomial theory), which we call pseudo-quantum Lakshmibai-Seshadri (LS) paths. This model carries less information than the alcove walks in the Ram-Yip formula, and it is therefore more efficient. Furthermore, we construct a connected pseudo-crystal structure on the pseudo-quantum LS paths, which is expected to lead to a simple Littlewood-Richardson rule for multiplying Macdonald polynomials. By contrast with the Kashiwara crystals, our pseudo-crystals have edges labeled by arbitrary roots.

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arXiv2018

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

Let $λ$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(λ)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $λ$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w λ}(q, t)$ are explicitly described in terms of QLS paths of shape $λ$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $λ$, $μ$, we have an isomorphism $Θ: \mathrm{QLS}(λ+ μ) \rightarrow \mathrm{QLS}(λ) \otimes \mathrm{QLS}(μ)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $Θ$ 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.

math.QA↗

arXiv2018

Generalized Weyl modules and Demazure submodules of level-zero extremal weight modules

We study a relationship between the graded characters of generalized Weyl modules $W_{w λ}$, $w \in W$, over the positive part of the affine Lie algebra and those of specific quotients $V_{w}^- (λ) / X_{w}^- (λ)$, $w \in W$, of the Demazure submodules $V_{w}^- (λ)$ of the extremal weight modules $V(λ)$ over the quantum affine algebra, where $W$ is the finite Weyl group and $λ$ is a dominant weight. More precisely, we prove that a specific quotient of the Demazure submodule is a quantum analog of a generalized Weyl module.

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arXiv2016

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 $P_λ^{A_{2n}^{(2)}} (q,0)$ of the symmetric Macdonald polynomial $P_λ^{A_{2n}^{(2)}} (q,t)$ at $t=0$, where $λ$ is a dominant weight and $P_λ^{A_{2n}^{(2)}}(q,t)$ denotes the specific specialization of the symmetric Macdonald-Koornwinder polynomial $P_λ(q,t_1, t_2, t_3, t_4, t_5)$. More precisely, as some results for untwisted affine types, the set of all ($A_{2n}^{(2)}$-type) quantum Lakshmibai-Seshadri paths of shape $λ$, 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 $P_λ^{A_{2n}^{(2)}} (q,0)$ of the symmetric Macdonald-Koornwinder polynomial.

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arXiv2016

Specialization of nonsymmetric Macdonald polynomials at $t=\infty$ and Demazure submodules of level-zero extremal weight modules

In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (λ)$ of the level-zero extremal weight module $V(λ)$ over a quantum affine algebra of an arbitrary untwisted type, here, $λ$ is a dominant integral weight, and $w_{\circ}$ denotes the longest element in the finite Weyl group $W$. Also, for each $x \in W$, we obtain a combinatorial formula for the specialization $E_{x λ} (q, \infty)$ at $t=\infty$ of the nonsymmetric Macdonald polynomial $E_{x λ} (q,t)$, and also one for the graded character $\mathrm{gch} V_{x}^- (λ)$ of the Demazure submodule $V_{x}^- (λ)$ of $V(λ)$, both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape $λ$.

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