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Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups

Amit Hazi

TL;DR

This work develops a diagrammatic framework for affine Weyl groups in characteristic $p>0$ and introduces a Frobenius-driven matrix recursion that categorifies a recursion on the group ring $\mathbb{Z}W$ through a matrix category of smaller objects. Central to the approach is a Frobenius functor $\Psi$ that embeds the diagrammatic Hecke category and its $F$-twist into a valuation-theoretic, hat{R}-extended matrix setting, yielding a concrete realization of the matrix recursion on Grothendieck groups. Decategorification provides a new lower bound for the $p$-canonical basis and, via the $p$-canonical tilting character formula, produces recursive lower bounds for characters of indecomposable tilting modules. The framework reveals a $p$-characteristic fractal structure within the affine Hecke category and bridges modular representation theory with the Lusztig-Williamson tilting/billiards program, offering computational and conceptual tools for exploring $p$-canonical phenomena.

Abstract

Let $W$ be an affine Weyl group, and let $\Bbbk$ be a field of characteristic $p>0$. The diagrammatic Hecke category $\mathcal{D}$ for $W$ over $\Bbbk$ is a categorification of the Hecke algebra for $W$ with rich connections to modular representation theory. We explicitly construct a functor from $\mathcal{D}$ to a matrix category which categorifies a recursive representation $ξ: \mathbb{Z}W \rightarrow M_{p^r}(\mathbb{Z}W)$, where $r$ is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are ``smaller'' by a factor of $p$. It also explains the presence of self-similarity in the $p$-canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the $p$-canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent $p$-canonical tilting character formula due to Achar-Makisumi-Riche-Williamson.

Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups

TL;DR

This work develops a diagrammatic framework for affine Weyl groups in characteristic and introduces a Frobenius-driven matrix recursion that categorifies a recursion on the group ring through a matrix category of smaller objects. Central to the approach is a Frobenius functor that embeds the diagrammatic Hecke category and its -twist into a valuation-theoretic, hat{R}-extended matrix setting, yielding a concrete realization of the matrix recursion on Grothendieck groups. Decategorification provides a new lower bound for the -canonical basis and, via the -canonical tilting character formula, produces recursive lower bounds for characters of indecomposable tilting modules. The framework reveals a -characteristic fractal structure within the affine Hecke category and bridges modular representation theory with the Lusztig-Williamson tilting/billiards program, offering computational and conceptual tools for exploring -canonical phenomena.

Abstract

Let be an affine Weyl group, and let be a field of characteristic . The diagrammatic Hecke category for over is a categorification of the Hecke algebra for with rich connections to modular representation theory. We explicitly construct a functor from to a matrix category which categorifies a recursive representation , where is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are ``smaller'' by a factor of . It also explains the presence of self-similarity in the -canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the -canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent -canonical tilting character formula due to Achar-Makisumi-Riche-Williamson.

Paper Structure

This paper contains 26 sections, 54 theorems, 253 equations, 12 figures.

Table of Contents

  1. Affine Weyl groups
  2. Affine Cartan matrices and realizations
  3. The Frobenius map
  4. Hecke algebras
  5. Ordinary Hecke algebras
  6. H_l|* and H_*
  7. Twisted l-defects and patterns
  8. Diagrammatic Soergel bimodules
  9. The diagrammatic Hecke category
  10. Light leaves
  11. Mixed diagrams and localization
  12. Dual-valued scalars
  13. The Frobenius functor
  14. A new ring
  15. One-color generators
  16. ...and 11 more sections

Key Result

Theorem I

There is a faithful monoidal functor which induces the matrix recursion representation on Grothendieck rings.

Figures (12)

  • Figure 1: The $l$-alcoves for $l=3$ and $W$ of type $\widetilde{A}_2$.
  • Figure 2: Examples of ${\rm dot}^{(F)}_s$ and ${\rm fork}^{(F)}_s$ ($p=3$, $W$ of type $\widetilde{A_2}$).
  • Figure 3: Examples of $\rho_{\tilde{s}}^{\underline{w},\tilde{s}'}$ and $\rho_{s}^{\underline{w},s'}$ for $p=3$ in type $\widetilde{A_2}$.
  • Figure 4: Examples of $\rho_{\tilde{s}}^{\underline{w},\tilde{s}'}$ and $\rho_{s}^{\underline{w},s'}$ for $p=3$ in type $\widetilde{C_2}$.
  • Figure 5: The morphism ${\rm braid}^{(F)}_{1,0}$ for $p=3$ in type $\widetilde{A_2}$.
  • ...and 7 more figures

Theorems & Definitions (157)

  • Theorem I
  • Theorem II
  • Conjecture : Andersen
  • Theorem : Recursive tilting character lower bound, à la Andersen
  • Theorem III
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • ...and 147 more