Related papers: A generalization of quantum Lakshmibai-Seshadri pa…

In this paper, we study level-zero extremal weight modules over twisted quantum affine algebras. To this end, we introduce semi-infinite Lakshmibai--Seshadri paths associated with a level-zero dominant integral weight $\lambda$. We then…

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…

We introduce semi-infinite Lakshmibai-Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an…

We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of…

We give interpretations of energy functions and (classically restricted) one-dimensional sums associated to tensor products of level-zero fundamental representations of quantum affine algebras in terms of Lakshmibai-Seshadri paths of…

We continue the study, begun in [Kouno-Naito-Orr-Sagaki, 2021], of inverse Chevalley formulas for the equivariant $K$-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial…

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…

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in…

In this paper, we propose an algebraic approach via Lakshmibai-Seshadri (LS) algebras to establish a link between standard monomial theories, Newton-Okounkov bodies and valuations. This is applied to Schubert varieties, where this approach…

Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant K-theory of a Kac-Moody flag manifold. The weight of the line bundle may be dominant…

We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula…

We give a Chevalley formula for an arbitrary weight for the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an…

Let $\lambda = \sum_{i \in I_{0}} m_{i} \varpi_{i}$, with $m_{i} \in \mathbb{Z}_{\ge 0}$ for $i \in I_{0}$, be a level-zero dominant integral weight for an affine Lie algebra $\mathfrak{g}$ over $\mathbb{Q}$, where the $\varpi_{i}$, $i \in…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set $\lambda: = \Lambda_1 - \Lambda_2$, where $\Lambda_1, \Lambda_2$ are the fundamental weights for $\mathfrak{g}$; note that $\lambda$ is neither dominant nor…

We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert…

Forgetting a subspace from a partial flag yields another partial flag composed of fewer subspaces. This induces a forgetful map $\pi : X \to X'$ between the corresponding flag varieties. We prove here that, for a degree large enough, the…

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a…

In this work, we introduce a Distributed Quantum Long Short-Term Memory (QLSTM) framework that leverages modular quantum computing to address scalability challenges on Noisy Intermediate-Scale Quantum (NISQ) devices. By embedding…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $\lambda$ be an arbitrary integral weight. We denote by $\mathbb{B}(\lambda)$ the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. Let $V(\lambda)$ be the…

We derive cancellation-free Chevalley-type multiplication formulas in the T-equivariant quantum K-theory of Grassmannians of type A and C, and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley…