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A generalization of quantum Lakshmibai-Seshadri paths for arbitrary weights

Takafumi Kouno, Satoshi Naito

Abstract

We construct an injective weight-preserving map (called the forgetful map) from the set of all admissible subsets in the quantum alcove model associated to an arbitrary weight. The image of this forgetful map can be explicitly described by introducing the notion of "interpolated quantum Lakshmibai-Seshadri (QLS for short) paths", which can be thought of as a generalization of quantum Lakshmibai-Seshadri paths. As an application, we reformulate, in terms of interpolated QLS paths, an identity of Chevalley type for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra, which is a representation-theoretic analog of the Chevalley formula for the torus-equivariant $K$-group of a semi-infinite flag manifold.

A generalization of quantum Lakshmibai-Seshadri paths for arbitrary weights

Abstract

We construct an injective weight-preserving map (called the forgetful map) from the set of all admissible subsets in the quantum alcove model associated to an arbitrary weight. The image of this forgetful map can be explicitly described by introducing the notion of "interpolated quantum Lakshmibai-Seshadri (QLS for short) paths", which can be thought of as a generalization of quantum Lakshmibai-Seshadri paths. As an application, we reformulate, in terms of interpolated QLS paths, an identity of Chevalley type for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra, which is a representation-theoretic analog of the Chevalley formula for the torus-equivariant -group of a semi-infinite flag manifold.
Paper Structure (22 sections, 23 theorems, 116 equations, 4 tables)

This paper contains 22 sections, 23 theorems, 116 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Basic notation and definitions
  3. Notation for root systems
  4. Affine root systems
  5. The quantum Bruhat graph
  6. Some specific reflection orders
  7. The quantum alcove model
  8. Alcove paths and admissible subsets
  9. Reduced chains of roots
  10. Interpolated QLS paths
  11. Integrality conditions
  12. Definition of interpolated QLS paths
  13. Interpolated LS paths
  14. The forgetful map
  15. Construction of the forgetful map
  16. ...and 7 more sections

Key Result

Theorem A

Let $\lambda \in P$ be an arbitrary weight and $w \in W$. There exists an injective weight-preserving map $\widetilde{\Xi}: \mathcal{A}(w, \Gamma_{\vartriangleleft}(\lambda)) \rightarrow \mathop{\mathrm{IQLS}}\nolimits(\lambda) \times W$.

Theorems & Definitions (72)

  • Theorem A: = Definition \ref{['def:forgetful']} + Theorem \ref{['thm:inj_forgetful']} + Proposition \ref{['prop:forgetful_wt']}
  • Theorem B: = Theorem \ref{['thm:im_forgetful']}
  • Theorem C: = Theorem \ref{['thm:gch_Chevalley']}
  • Definition 2.1: BFP
  • Definition 2.2: cf. BB
  • Definition 2.3: cf. D
  • Remark 2.4
  • Lemma 2.5
  • Corollary 2.6
  • proof
  • ...and 62 more