Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hom formula for Soergel modules

Leonardo Patimo

TL;DR

The work extends Soergel bimodules to arbitrary Coxeter groups and unveils a refined module-theoretic picture: a distinguished submodule $H_w$ inside each indecomposable $B_w$ behaves like a cohomology submodule, and two independent constructions (via the structure algebra $Z$ and via light leaves) yield a coherent framework for $H_w$. By passing to the module category over the structure algebra and using a robust hom-formula, the authors show that morphism spaces between Soergel modules encode the Hecke algebra pairing, i.e., $\mathrm{grdim}\,\mathrm{Hom}^\bullet_{\overline Z}(\overline B,\overline B')=(\overline{[B]},[B'])$, generalizing Soergel’s categorification results. They also establish a translation-functor formalism on $Z$-modules, enabling a structural approach to morphisms and inductions. Crucially, the paper provides counterexamples showing that naive Hom-formulas over $R$ fail for infinite $W$, both in affine and universal Coxeter cases, highlighting the necessity of the structure-algebra perspective. Together, these results link algebraic, combinatorial, and (where available) geometric data to extend top-heaviness and Hodge-type phenomena to arbitrary Coxeter groups, and illuminate the limits of straight $R$-module Homomorphism approaches in infinite settings.

Abstract

We study Soergel modules for arbitrary Coxeter groups. For infinite Coxeter groups, we show that the homomorphisms between Soergel modules are in general more than those coming from morphisms of Soergel bimodules. This result provides a negative answer to a question posed by Soergel. We further show that the dimensions of the morphism spaces agree with the pairing in the Hecke algebra when Soergel modules are instead regarded as modules over the structure algebra. Moreover, we use this module structure to define a distinguished submodule of indecomposable Soergel bimodules that mimics the cohomology submodule of the intersection cohomology. Combined with the Hodge theory of Soergel bimodules, this can be used to extend results regarding the shape of Bruhat intervals, such as top-heaviness, to arbitrary Coxeter groups.

A Hom formula for Soergel modules

TL;DR

The work extends Soergel bimodules to arbitrary Coxeter groups and unveils a refined module-theoretic picture: a distinguished submodule inside each indecomposable behaves like a cohomology submodule, and two independent constructions (via the structure algebra and via light leaves) yield a coherent framework for . By passing to the module category over the structure algebra and using a robust hom-formula, the authors show that morphism spaces between Soergel modules encode the Hecke algebra pairing, i.e., , generalizing Soergel’s categorification results. They also establish a translation-functor formalism on -modules, enabling a structural approach to morphisms and inductions. Crucially, the paper provides counterexamples showing that naive Hom-formulas over fail for infinite , both in affine and universal Coxeter cases, highlighting the necessity of the structure-algebra perspective. Together, these results link algebraic, combinatorial, and (where available) geometric data to extend top-heaviness and Hodge-type phenomena to arbitrary Coxeter groups, and illuminate the limits of straight -module Homomorphism approaches in infinite settings.

Abstract

We study Soergel modules for arbitrary Coxeter groups. For infinite Coxeter groups, we show that the homomorphisms between Soergel modules are in general more than those coming from morphisms of Soergel bimodules. This result provides a negative answer to a question posed by Soergel. We further show that the dimensions of the morphism spaces agree with the pairing in the Hecke algebra when Soergel modules are instead regarded as modules over the structure algebra. Moreover, we use this module structure to define a distinguished submodule of indecomposable Soergel bimodules that mimics the cohomology submodule of the intersection cohomology. Combined with the Hodge theory of Soergel bimodules, this can be used to extend results regarding the shape of Bruhat intervals, such as top-heaviness, to arbitrary Coxeter groups.

Paper Structure

This paper contains 14 sections, 26 theorems, 76 equations.

Table of Contents

  1. Background and notation
  2. The structure algebra of a Coxeter group
  3. The nil Hecke ring and its dual
  4. Moment graphs of Coxeter groups
  5. Relationship with Soergel bimodules
  6. Cohomology and Homology within Soergel bimodules
  7. Light leaves basis
  8. Invariant forms and duality of Soergel bimodules
  9. Support filtration
  10. Cohomology
  11. Translation functors on Z-mod
  12. Counterexamples
  13. A counterexample in the affine Weyl group of type A??
  14. A counterexample in the universal Coxeter group of rank 3

Key Result

Theorem A

The indecomposable Soergel bimodule $B_w$ has a submodule $H_w$ which is free as a left (or right) $R$-module with basis $\{\mathcal{P}_x\}_{x\leq w}$. In particular, its graded rank over $R$ is

Theorems & Definitions (67)

  • Theorem A
  • Theorem B: Soergel's hom formula for Soergel modules
  • Definition 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Definition 2.1
  • ...and 57 more