[Paper Review] Sur la catégorie des bimodules de Soergel

Nicolás Libedinsky|ArXiv.org|Jul 24, 2007

Algebraic structures and combinatorial models4 references4 citations

TL;DR

This paper provides a combinatorial description of morphism spaces in Soergel's bimodule category $\mathbf{B}$ associated to a Coxeter system $(W,\mathcal{S})$, establishing a basis—called the 'light leaves basis'—for these morphism spaces as right modules over a polynomial ring. The key contribution is proving that this basis is canonical and explicitly computes graded dimensions, leading to an explicit description of morphisms in the principal block $\mathcal{O}_0$-proj of BGG category $\mathcal{O}$, via Soergel's equivalence.

ABSTRACT

The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La catégorie B de Soergel d'un système de Coxeter (W,S) est une catégorie de bimodules sur une algèbre de polynômes sur laquelle W agit. C'est une catégorification de l'algèbre de Hecke de (W,S). Dans cet article nous donnons une description combinatoire des espaces de morphismes dans B. En corollaire, on obtient une description analogue des morphismes dans O_0-proj, où O_0 est le bloc principal de la catégorie O de BGG.

Motivation & Objective

• To provide a combinatorial, explicit description of the morphism spaces in Soergel's bimodule category $\mathbf{B}$ associated to a Coxeter system $(W,\mathcal{S})$.
• To construct and prove the existence of a canonical basis—called the 'light leaves basis'—for the morphism spaces in $\mathbf{B}$, viewed as right modules over a polynomial ring.
• To apply this basis to explicitly compute morphisms in the principal block $\mathcal{O}_0$-proj of BGG category $\mathcal{O}$, using Soergel's equivalence between $\mathcal{O}_0$-proj and the category $\mathbf{B}^\mathbb{C}$.

Proposed method

• Define the category $\mathbf{B}$ as the category of Soergel bimodules over a polynomial ring $R$, with $W$-action via a faithful reflection representation.
• Introduce the 'light leaves basis' (BFL) as a recursive construction of morphisms in $\mathbf{B}$, mimicking the recurrence of the product $(1+T_{s_1})\cdots(1+T_{s_n})$ in the Hecke algebra.
• Use a recursive inductive construction on reduced expressions in $W$, defining morphisms via tangle-like relations and dual bases.
• Leverage Soergel's theorem that $\mathbf{B}$ categorifies the Hecke algebra $\mathcal{H}$, and that the Grothendieck group of $\mathbf{B}$ is isomorphic to $\mathcal{H}$ via the map $\varepsilon$.
• Establish an explicit isomorphism between the morphism spaces in $\mathbf{B}$ and the coinvariant algebra $C = R/R_+^W$, using the canonical map $\mathrm{Hom}_{(R,R)}(B,B') \otimes_R \mathbb{C} \simeq \mathrm{Hom}_C(B\otimes_R \mathbb{C}, B'\otimes_R \mathbb{C})$.
• Prove that the light leaves basis forms a free basis for $\mathrm{Hom}_{(R,R)}(B,B')$ as a right $R$-module, using the graded dimension formula from Corollary 4.2.

Experimental results

Research questions

• RQ1 Can the morphism spaces in Soergel's bimodule category $\mathbf{B}$ be described combinatorially in terms of a canonical basis?
• RQ2 Does the light leaves basis constructed via recursive tangle-like relations form a free basis for the morphism spaces in $\mathbf{B}$ as right $R$-modules?
• RQ3 Can the combinatorial description of morphisms in $\mathbf{B}$ be used to explicitly compute morphisms in the principal block $\mathcal{O}_0$-proj of BGG category $\mathcal{O}$?

Key findings

• The light leaves basis (BFL) is proven to be a free basis for the morphism spaces $\mathrm{Hom}_{(R,R)}(B,B')$ in the category $\mathbf{B}$, as a right module over the polynomial ring $R$.
• The graded dimensions of the morphism spaces in $\mathbf{B}$ are given by the coefficient of $1$ in the product $(1+T_{s_1})\cdots(1+T_{s_n})$, as stated in Corollary 4.2.
• The construction of the BFL is explicitly recursive and mimics the recurrence of the product in the Hecke algebra, providing a categorification of this algebraic formula.
• The morphism spaces in the category $\mathcal{O}_0$-proj are explicitly described via the isomorphism $\mathrm{Hom}_{(R,R)}(B,B') \otimes_R \mathbb{C} \simeq \mathrm{Hom}_C(B\otimes_R \mathbb{C}, B'\otimes_R \mathbb{C})$, where $C = R/R_+^W$ is the coinvariant algebra.
• The basis elements in $\mathrm{Hom}_{(R,R)}(B,B')$ are constructed via a map involving $x_{\overline{i}}^g f_\alpha((1\otimes x_{\overline{i}^{c}(\mathrm{op})}^d)\cdot m)$, forming a basis over $R$.
• The result implies that the structure of morphisms in $\mathcal{O}_0$-proj is completely determined by the combinatorics of the BFL in $\mathbf{B}$, providing a new explicit computational tool.