Exotic Grading in Special Characteristic

• Exotic grading in special characteristic is a refined grading structure on the Hecke category, incorporating a Z ⊕ p-torsion component that distinguishes Frobenius shifts from the standard grading.
• It enables the categorification of Hecke algebras with unequal parameters by aligning exotic degree assignments with Frobenius actions and intrinsic symmetry shifts.
• This structure has practical implications in representation theory and derived category analysis, producing new Koszul gradings and structural insights in modular settings.

Exotic grading in special characteristic refers to a distinguished refinement of the standard grading structure on the Hecke (Soergel) category when it is defined over a field of positive characteristic. This grading, which is intrinsically linked to the action of the Frobenius automorphism and the presence of $p$-torsion in the fundamental group of the dual root system, enables a categorification of Hecke algebras with unequal parameters and introduces new structural symmetries absent in characteristic zero. It has significant consequences in representation theory, category theory, and the study of algebraic groups in modular contexts.

1. Grading Structures on the Hecke Category

Let $(W, S)$ denote a finite or affine Coxeter system attached to a crystallographic root datum, and let $\Bbbk$ be a field of prime characteristic $p$. The Hecke category $\mathcal{H}_C$ is realized as the graded category of Soergel bimodules over $R = \operatorname{Sym}(\mathfrak{h}^*)$, with the standard integer grading assigned as $\deg_{\mathrm{std}}(x) = 2$ for all $x \in \mathfrak{h}^*$. The indecomposable objects are Bott–Samelson bimodules $B_{s_1\ldots s_m}$, and morphism spaces are graded by the usual integer degree.

The presence of a Frobenius endomorphism $\operatorname{Fr}^*: R \to R$, $f(x) \mapsto f(x)^p$, determined by the characteristic $p$, induces a Frobenius–pullback monoidal autoequivalence $\operatorname{Fr}: \mathcal{H}_C \to \mathcal{H}_C$. This functor is grading-preserving up to shift and commutes with degree shifts $\langle 1 \rangle$ of the standard grading, satisfying $\operatorname{Fr}(B_s) \cong B_s$ on objects (Antor et al., 29 Dec 2025).

2. The Exotic Grading Group and Assignment

The exotic grading is defined in the context where the $p$-torsion subgroup $\Pi[p]$ of the fundamental group $\Pi = \Lambda^\vee/\Lambda_r^\vee$ of the dual root system is nontrivial. The exotic grading group is then

$\Gamma = \mathbb{Z} \oplus \Pi[p]$

with elements denoted $(i, \bar\mu)$. The $\Gamma$-grading on $\mathcal{H}_C$ is a decomposition: $$\operatorname{Hom}_{\mathcal{H}_C}(M, N) = \bigoplus_{(i, \bar\mu) \in \Gamma} \operatorname{Hom}^{i, \bar\mu}(M, N)$$ This grading satisfies:

• The projection $\Gamma \to \mathbb{Z}$ recovers the standard grading.
• Composition and monoidal products add bidegrees in $\Gamma$.
• The Frobenius functor $\operatorname{Fr}$ is homogeneous of degree $(0, \bar\omega)$, for a generator $\bar\omega \in \Pi[p]$.

Assignments for generators:

• $\deg_{\mathrm{ex}}(B_s) = (1, \bar\omega_s)$, where $\bar\omega_s$ is the $p$-torsion image of the simple coroot $\alpha_s^\vee$.
• $\deg_{\mathrm{ex}}(x : R \to R) = (2, 0)$ for $x \in \mathfrak{h}^*$.
• Shifts: $\langle 1 \rangle$ corresponds to $(1, 0)$, and $\operatorname{Fr}$ to $(0, 1)$ in $\mathbb{Z} \oplus \Pi[p]$.

For Bott–Samelson objects,

$$\deg_{\mathrm{ex}}(B_{s_1\ldots s_m}) = \left(m, \sum_{j=1}^m \bar\omega_{s_j}\right)$$

(Antor et al., 29 Dec 2025).

3. Classification of Refinement Gradings

The classification theorem for refinements of the standard grading establishes that the only nontrivial new grading on the Hecke category, besides scalar extensions, arises as the exotic grading in special characteristic. There is a bijection between such gradings and group homomorphisms $\phi: \Pi \to \Gamma$ subject to the image of $1 \in \mathbb{Z} \subset \Gamma$ corresponding to the grading shift. If the characteristic $p$ does not divide $|\Pi|$, all refinements are trivial. Otherwise, when $p$ divides the order, the unique (up to equivalence) exotic grading is the only extension: $$\Pi[p] \hookrightarrow \Gamma = \mathbb{Z} \oplus \Pi[p]$$ This structure governs the possible grading data and autoequivalences of $\mathcal{H}_C$ beyond the standard grading (Antor et al., 29 Dec 2025).

4. Frobenius Automorphism and Grading Symmetries

The Frobenius autoequivalence $\operatorname{Fr}$ acts as a symmetry of the Hecke category, shifting only the $\Pi[p]$-component in the exotic grading. Explicitly,

$$\deg_{\mathrm{ex}}(\operatorname{Fr}(f)) = (0, 1) + \deg_{\mathrm{ex}}(f)$$

for any morphism $f$, whereas the ordinary grading shift acts as

$$\deg_{\mathrm{ex}}(f \langle 1 \rangle) = (1, 0) + \deg_{\mathrm{ex}}(f)$$

The commutativity of these operations generates a subgroup $$\mathbb{Z}^2 \subset \operatorname{End}(\mathcal{H}_C)$$, giving rise to the entire exotic grading structure. All monoidal autoequivalences of $\mathcal{H}_C$ that fix isomorphism classes of Bott–Samelson bimodules are compositions of grading shifts and the Frobenius functor, reflecting that the exotic shift is the only truly new symmetry in special characteristic (Antor et al., 29 Dec 2025).

5. Categorification of Unequal-Parameter Hecke Algebras

A key application of the exotic grading is to categorify Hecke algebras $\mathcal{H}_q(W, S; \{v_s\})$ with unequal parameters. Via the function

$m: S \to \mathbb{Z}/(p-1)$

with $v_s = v^{1 + m(s)}$, and $m$ factoring through $S \hookrightarrow \Pi[p] \cong \mathbb{Z}/(p-1)$, the assignment of exotic degrees to generators ensures that the Grothendieck group $K_0^\Gamma(\mathcal{H}_C)$ is a free $\mathbb{Z}[v^{\pm 1}]$-module on the classes $[B_w]$, and that the induced relations recover the unequal-parameter Hecke algebra: $$[B_s] \mapsto T_s + v_s^{-1}, \qquad v \mapsto \langle 1 \rangle$$ All braid and quadratic relations are respected in $K_0^\Gamma(\mathcal{H}_C)$, thereby providing a categorification framework for all finite and affine Weyl groups with arbitrary parameter assignments when $p$-torsion is present (Antor et al., 29 Dec 2025).

6. Concrete Examples of Exotic Grading

Two representative cases illustrate the construction:

Type	$p$	$\Pi$	$\Pi[p]$	$\Gamma$	$\deg_{\mathrm{ex}}$ assignments
$A_2$	3	$\mathbb{Z}/3$	$\mathbb{Z}/3$	$\mathbb{Z} \oplus \mathbb{Z}/3$	$B_1: (1,1),\ B_2: (1,2)$
$\widetilde{A}_1$	2	$\mathbb{Z}/2$	$\mathbb{Z}/2$	$\mathbb{Z} \oplus \mathbb{Z}/2$	$B_{s_0}, B_{s_1}: (1,1)$

For $A_2$, one obtains the $(v, v^2)$-unequal-parameter Hecke algebra, corresponding to the possible assignments of exotic grading. For affine rank one $\widetilde{A}_1$ at $p=2$, the structure allows both the standard equal-parameter setting (when the exotic shifts coincide) and a non-standard unequal-parameter version by splitting the grading on the generators (Antor et al., 29 Dec 2025).

7. Interactions with Derived Categories and Koszulity

The exotic grading has implications beyond the additive category. In derived or mixed-derived settings (e.g., the hearts of exotic $t$-structures on derived categories of equivariant coherent sheaves), it refines the usual mixed grading to an $E_2$-structure that often produces new Koszul gradings on blocks of modular category $\mathcal{O}$ in characteristic $p$. The corresponding tilting objects, standard and costandard objects, and their filtrations realize the structure of a graded highest-weight category, with the geometric braid group action providing precise combinatorial control over gradings and extensions (Mautner et al., 2014). This grading is independent of the characteristic for the existence and combinatorial structure of the exotic heart, but identification of certain tilting objects with $G$-tilting modules requires favorable (good) characteristic.

In summary, exotic grading in special characteristic uniquely enriches the grading structure of the Hecke category when $p$-torsion is present in the dual root system, intertwining the standard grading and Frobenius symmetries, and enabling categorification of Hecke algebras with arbitrary parameter sets. The construction is robust, with repercussions in the representation theory of algebraic groups, the structure of derived categories, and the classification of category autoequivalences (Antor et al., 29 Dec 2025, Mautner et al., 2014).