Singular K-Theory Soergel Bimodules Overview

• Singular K-theory Soergel bimodules are bigraded modules modeling the equivariant algebraic K-theory of parabolic Bott–Samelson resolutions and categorifying Hecke algebras.
• They are constructed using parabolic and singular settings with diagrammatic and combinatorial presentations governed by quadratic and braid relations.
• They connect geometric, algebraic, and quantum frameworks, linking Ext-groups, Grothendieck groups, and quantum Satake equivalences.

Singular K-theory Soergel bimodules are a class of bigraded bimodules arising as categorical models for the equivariant algebraic K-theory of parabolic Bott–Samelson resolutions. They play a central role in the categorification of K-theoretic Hecke algebras and provide a foundational language for describing the structure and actions in parabolic and singular categorical settings, including quantum and motivic contexts. These bimodules interpolate between diagrammatic, algebraic, and geometric incarnations of Hecke-theoretic representation theory, organizing both Ext-groups and Grothendieck groups, and inducing deep connections with link homologies and quantum Satake equivalences (Eberhardt et al., 24 Nov 2025, Williamson, 2010, Khovanov et al., 2016).

1. Definition and Geometric Realization

Given a complex connected reductive group $G$ with maximal torus $T \subset B \subset G$ and associated Weyl group $W$ with simple reflections $S$, the framework of singular K-theory Soergel bimodules is grounded in equivariant algebraic K-theory. For each subset $I \subset S$, let $P_I$ denote the corresponding parabolic subgroup and set

$$R = K^0_T(\mathrm{pt}) = \operatorname{Rep}(T)[v,v^{-1}], \qquad R^I = K^0_T(G/P_I) \cong (\operatorname{Rep}(T))^{W_I}[v,v^{-1}].$$

For words $\uw = (s_{i_1},\ldots,s_{i_\ell})$ in the simple reflections, one defines the parabolic Bott–Samelson resolution $\mathrm{BS}(\uw)$ and the associated proper $T$–equivariant map

$\mathrm{BS}_J^I(\uw) \to G/P_J \times G/P_I.$

The singular K-theory Soergel bimodule corresponding to $\uw$ is

$\bim(\uw) := K^0_T(\mathrm{BS}_J^I(\uw)) \in (R^J\text{-Mod})\text{-}(R^I\text{-Mod}),$

yielding an $(R^J, R^I)$-bimodule. For the generating simple reflections $s \in S$, the basic bimodules are given by

$$B_s^I = R^I \otimes_{R^{I \cup \{s\}}} R^I, \qquad {^J\!B_s} = R^J \otimes_{R^{J\cup\{s\}}} R^J.$$

These data produce a combinatorial model for the equivariant K-theory of G/P double cosets and their Bott–Samelson resolutions (Eberhardt et al., 24 Nov 2025).

2. Diagrammatic and Combinatorial Presentation

Singular K-theory Soergel bimodules admit a diagrammatic and algebraic presentation generalizing the Coxeter-type Hecke algebra categorification. The category $\mathsf{SBim}$ is defined as the smallest full subcategory of bigraded $(R^I, R^J)$-bimodules closed under finite direct sums, Bott (cohomological) shifts $[1]$ (multiplication by $v$), weight-gradings $\langle 1 \rangle$, tensoring over common invariants, and direct summands (Karoubi envelope). The set of generators includes

• Simple reflection bimodules $B_s^I$ for all $I \subset S$ and $s \notin I$,
• Trivial bimodules $R^I$.

Relations are governed by K-theoretic upgrades of the canonical quadratic and braid relations:

• Quadratic: $$B_s^I \otimes_{R^I} B_s^I \cong B_s^I\langle 1 \rangle \oplus B_s^I[-1]\langle -1 \rangle$$
• Braid: For $s, t \in S$ with braid relation of length $m$, tensor powers of $B_s$ and $B_t$ satisfy the usual braid group isomorphism for $m$ factors.

Convolution corresponds to tensor product over the intermediate invariants. Hom spaces have natural adjunctions inherited from the bimodule structure, compatible with both gradings (Eberhardt et al., 24 Nov 2025).

3. Grading Structures and Grothendieck Groups

The category is naturally $\mathbb{Z} \oplus \mathbb{Z}$-graded. The two gradings are distinguished as follows:

• The cohomological ('Bott') grading: shift by $[1]$ multiplies by $v$,
• The weight grading: shift by $\langle 1 \rangle$ corresponds to twisting by a one-dimensional character of $T$.

For any object $M$, its bigraded dimension

$$\mathrm{grdim}_{v, t}(M) = \sum_{a,b \in \mathbb{Z}} \left(\dim \operatorname{Hom}(R^I, M[a]\langle b\rangle)\right) v^a t^b$$

packages its K-theoretic and torus-weight information. The Grothendieck group $K_0(\mathsf{SBim})$ recovers a categorified version of the parabolic K-theoretic Hecke algebra $\mathcal{H}_I^K$, with classes $[B_s^I]$ matching the standard generators $T_s$ (Eberhardt et al., 24 Nov 2025). For the original singular Soergel categories, the split Grothendieck group is isomorphic to the Schur algebroid, with explicit basis given by parabolic Kazhdan–Lusztig elements (Williamson, 2010).

4. Categorical and Braid Group Actions

In specific ranks and types, e.g., for type $A$ and $k=1$, the diagrammatic algebras $W(n,1)$ (the "redotted Webster algebras") act as endomorphism algebras of direct sums of indecomposable singular Soergel bimodules. A categorical braid group action is constructed via explicit two-term complexes (Rouquier complexes) of bimodules: $$\sigma_i = [W \to W_i\langle -1 \rangle], \qquad \sigma_i^{-1} = [W_i\langle +1 \rangle \to W],$$ where $W_i$ involves partial tensor products corresponding to simple transpositions. These complexes satisfy the braid relations in the homotopy category $\mathrm{Kom}(W(n,1)\text{-gpmod})$ and induce a Burau representation on the split K-theory, categorifying the classical braid group action (Khovanov et al., 2016). Analogous constructions are conjectured for higher $k$ and general parabolic cases.

5. Classification and Character Theory

For arbitrary Coxeter systems and finitary subsets $I, J \subset S$, indecomposable singular Soergel bimodules are classified by double cosets $W_I \backslash W / W_J$, partially ordered by Bruhat order. Each class corresponds to a unique (up to shift) indecomposable object $B_p^{I,J}$, which is self-dual under graded duality. Induction and restriction functors are realized as categorical correspondences between these modules.

Every bimodule admits filtrations by standard (Δ) and costandard (∇) objects. The decategorified "character" of each indecomposable matches the self-dual standard basis in the parabolic Hecke algebra; in the non-singular case this is the Kazhdan–Lusztig basis. Soergel’s conjecture relates these characters directly to the canonical Hecke algebra classes, and its verification for the non-singular case implies analogous results for all singular bimodules (Williamson, 2010).

6. Connections to Parabolic Motives and Quantum Satake

The motivic incarnation of the singular K-theory Soergel category appears as a full subcategory in the category of $G$-equivariant mixed K-motives on products of partial flag varieties. The main result is an equivalence: $$\mathrm{DM}^K_{(P_I,P_J)}(G) \simeq \mathsf{Kar}\langle \{ B_s^I \mid s \notin I\}, \otimes_{R}, [1], \langle 1 \rangle \rangle,$$ relating geometric motives to diagrammatic generators and tensors. This equivalence enables the computation of K-theoretic Hecke algebra structures, connects diagrammatic presentations (such as the Elias–Khovanov and Webster formalisms) to topological and geometric realization, and provides categorical tools for quantum K-theoretic Satake correspondences (Eberhardt et al., 24 Nov 2025, Khovanov et al., 2016).

7. Illustrative Examples and Computational Tools

For $\mathrm{GL}_2$ (type $A_1$), the basic module structure is as follows:

• $R = \mathbb{Z}[t_1^{\pm1}, t_2^{\pm1}][v, v^{-1}]$,
• $$R^s = \mathbb{Z}[t_1^{\pm1} + t_2^{\pm1}][v, v^{-1}]$$.

The unique nontrivial simple bimodule $B_s = R \otimes_{R^s} R$ is free of rank 2 as an $R$-bimodule, and one verifies the quadratic relation directly. In the rank 2 case, the redotted Webster algebras $W(2,1)$ correspond precisely to the endomorphism algebras of the two singular Soergel bimodules, and their complexes recover the Burau representation on K-theory via explicit q-matrices (Eberhardt et al., 24 Nov 2025, Khovanov et al., 2016). These concrete models enable explicit computations of Ext-groups and decategorified invariants.

---

Key references: (Eberhardt et al., 24 Nov 2025, Williamson, 2010, Khovanov et al., 2016).