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Reduction rules for Demazure modules

Marc Besson, Sam Jeralds, Joshua Kiers

TL;DR

The paper proves a reduction rule for weight multiplicities in Demazure modules: if a weight $\\mu$ lies on a face $\\mathcal{F}(v,\\eta)$ of the Demazure weight polytope $P^w_\\lambda$, then the multiplicity is preserved under passage to a Levi-subgroup Demazure module, namely $\\text{dim} V^w_\\lambda(\\mu) = \\text{dim} V^{w_L}_{\\lambda_L}(\\mu)$, with explicit Levi data $L, w_L, \\\lambda_L$. The authors combine algebro-geometric arguments (Roth-type reductions via embeddings $i_{L,z}: L/B_L \to G/B$ and descent of line bundles) with convex-geometric and Demazure-operator analyses to relate the weight spaces on faces of $P^w_\\lambda$ to those in a Levi Demazure module. The approach mirrors classical reduction rules and uses the Hilbert–Mumford framework for ample cones, showing that faces of the ample cone correspond to Levi reductions. The results provide a concrete and computable bridge between $G$-level and Levi-level weight multiplicities, illuminating how extremal data encoded by faces controls the reduction. This has potential implications for branching problems and the combinatorics of Demazure characters via geometric and polyhedral methods.

Abstract

For $G$ a complex reductive group and $B \subseteq G$ a Borel subgroup, we provide a reduction rule for certain weight multiplicities in Demazure modules $V_λ^w$: given a weight $μ$ on a face of the associated weight polytope $P_λ^w$, we reduce the computation of the dimension of the weight space $V_λ^w(μ)$ to a similar problem of computing the weight space dimension for a Demazure module of a Levi subgroup of $G$.

Reduction rules for Demazure modules

TL;DR

The paper proves a reduction rule for weight multiplicities in Demazure modules: if a weight lies on a face of the Demazure weight polytope , then the multiplicity is preserved under passage to a Levi-subgroup Demazure module, namely , with explicit Levi data . The authors combine algebro-geometric arguments (Roth-type reductions via embeddings and descent of line bundles) with convex-geometric and Demazure-operator analyses to relate the weight spaces on faces of to those in a Levi Demazure module. The approach mirrors classical reduction rules and uses the Hilbert–Mumford framework for ample cones, showing that faces of the ample cone correspond to Levi reductions. The results provide a concrete and computable bridge between -level and Levi-level weight multiplicities, illuminating how extremal data encoded by faces controls the reduction. This has potential implications for branching problems and the combinatorics of Demazure characters via geometric and polyhedral methods.

Abstract

For a complex reductive group and a Borel subgroup, we provide a reduction rule for certain weight multiplicities in Demazure modules : given a weight on a face of the associated weight polytope , we reduce the computation of the dimension of the weight space to a similar problem of computing the weight space dimension for a Demazure module of a Levi subgroup of .
Paper Structure (19 sections, 26 theorems, 81 equations, 3 figures)

This paper contains 19 sections, 26 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Reduction Rules
  3. Ample cones and reduction rules
  4. Reduction rules for Demazure modules
  5. Outline of the paper
  6. Acknowledgements
  7. Notations and preliminaries on Demazure modules
  8. Notation for reductive groups
  9. Weyl group combinatorics
  10. Flag variety geometry
  11. Schubert Geometry and Demazure modules
  12. Weight polytopes for Demazure modules
  13. Faces and Inequalities of $P_\lambda^w$
  14. Proof of Theorem \ref{['PreciseTheorem']}(1)
  15. Embeddings of Schubert varieties
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mu$ be a weight of the Demazure module $V^w_\lambda$ with $\mu$ lying on the face $\mathcal{F}(v,\eta)$ of the associated polytope $P^w_\lambda$. Then where $L:= \dot{v}L_\eta \dot{v}^{-1}$ is the conjugate of the Levi $L_\eta$ determined by $\eta$, $w_L:= v \pi_\eta(w^{-1} \ast v)^{-1} v^{-1}$, $\lambda_L:=v \pi^{\eta}(w^{-1} \ast v) \lambda$, and $V^{w_L}_{\lambda_L}$ the Demazure module

Figures (3)

  • Figure 1: Demazure polytopes for type $B_3$, highest weight $\lambda = \rho = \omega_1+\omega_2+\omega_3$. The shaded region is for $q = s_3s_2s_3s_1s_2s_3$; the wire frame extends to the polytope for $w = s_1s_3s_2s_3s_1s_2s_3$. The face $\mathcal{F}(s_1,x_1)$ corresponding to $\eta = x_1, v = s_1$ is common to both polytopes and has been highlighted. The weight spaces on that face have been emphasized, where a single $\bullet$ stands for multiplicity one, and each concentric ring indicates an increase in multiplicity from there.
  • Figure 2: Demazure module of type $B_2$ with highest weight $2\omega_1 + \omega_2$ and Weyl group element $s_2s_1s_2$.
  • Figure 3: Demazure polytope for type $B_3$, highest weight $\lambda = \rho = \omega_1+\omega_2+\omega_3$ and $w = s_1s_3s_2s_3s_1s_2s_3$. Note that $s_1*w = w$. The face corresponding to $\eta = x_1, v = s_1$ is highlighted in orange, while the face corresponding to $\eta = x_1, v = e$ is highlighted in blue. The two faces and their weight spaces are in bijection via $s_1$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Corollary 3.7
  • ...and 41 more