Connectedness of Lakshmibai-Seshadri path crystals for hyperbolic Kac-Moody algebras of rank 2

Lakshmibai-Seshadri paths and non-symmetric Cauchy identity

We give a simple crystal theoretic interpretation of the Lascoux's expansion of a non-symmetric Cauchy kernel $\prod_{i+ j\leq n+1}(1-x_iy_j)^{-1}$, which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.

讲堂多面体与拉克希米巴伊-塞沙德里路径

Lecture hall polytopes and Lakshmibai-Seshadri paths

本文通过建立讲堂多面体格点与L-S路径的双射，完整证明了讲堂多面体的Koszul性质，并为其整数分解性质和Gorenstein性质判定准则提供了新证明。

任意权下量子Lakshmibai-Seshadri路径的推广

A generalization of quantum Lakshmibai-Seshadri paths for arbitrary weights

该研究构造了量子凹室模型可允许子集到插值QLS路径的保权单射，推广了量子Lakshmibai-Seshadri路径，并以此重述了量子仿射代数零级极值权模Demazure子模分次特征的Chevalley型恒等式。

Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $λ$ be an arbitrary integral weight. We denote by $\mathbb{B}(λ)$ the crystal of all Lakshmibai-Seshadri paths of shape $λ$. Let $V(λ)$ be the extremal weight module of extremal weight $λ$ generated by the (cyclic) extremal weight vector $v_λ$ of weight $λ$, and let $\mathcal{B}(λ)$ be the crystal basis of $V(λ)$ with $u_λ\in \mathcal{B}(λ)$ the element corresponding to $v_λ$. We prove that the connected component $\mathcal{B}_0(λ)$ of $\mathcal{B}(λ)$ containing $u_λ$ is isomorphic, as a crystal, to the connected compon…

Equivariant K-Chevalley Rules for Kac-Moody Flag Manifolds

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 or antidominant, and the coefficients may be described either by Lakshmibai-Seshadri paths or by the alcove model of the first author and Postnikov. For Lakshmibai-Seshadri paths, our formulas are the Kac-Moody generalizations of results of Griffeth-Ram and Pittie-Ram for finite dimensional flag manifolds. A gap in the proofs of the mentioned results is add…

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-Richardso…