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Decategorification of a Rouquier complex is the Hecke braid generator

Statement

Let $Φ ⁣ : K_{0}^{s p l i t} ({S B i m}_{n}) \to H_{S_{n}}$ be the unique algebra isomorphism of The split Grothendieck group of the Soergel category is the type-A Hecke algebra with $Φ ([B_{i}]) = H_{i} = v (T_{i} + 1)$ and $Φ (v X) = v Φ (X)$ , where $H_{S_{n}}$ is the Hecke algebra over $A = Z [v, v^{- 1}]$ with $q = v^{- 2}$ and standard generators $T_{i}$ , and let $χ$ be the alternating class of Euler classes of Rouquier complexes are homotopy invariant and multiplicative.

(a) $Φ (χ (F_{i})) = H_{i} - v = v T_{i}$ and $Φ (χ (F_{i}^{- 1})) = H_{i} - v^{- 1} = v^{- 1} T_{i}^{- 1}$ ; equivalently $χ (F_{i} (- 1))$ maps to $T_{i}$ and $χ (F_{i}^{- 1} (1))$ maps to $T_{i}^{- 1}$ . In particular $χ (F_{i}) χ (F_{i}^{- 1}) = 1$ , and the powers of $v$ are the exact normalization of the library's shift convention, not an ambiguity.

(b) For every braid $β \in B_{n}$ and every signed word $σ$ for $β$ , $Φ (χ (F (σ))) = v^{e (σ)} T_{β}$ , where $T_{β} : = T_{i_{1}}^{ϵ_{1}} \dots T_{i_{r}}^{ϵ_{r}}$ is the image of $β$ under the standard group homomorphism $B_{n} \to H_{S_{n}}^{\times}$ , $σ_{i} \mapsto T_{i}$ , and $e (σ) = # {ϵ_{k} = 1} - # {ϵ_{k} = - 1}$ depends only on $β$ . Hence $β \mapsto v^{- e (β)} χ (F (β))$ , composed with $Φ$ , is that standard homomorphism: up to the grading normalization, the Rouquier complex of a braid decategorifies to the image of the braid in the Hecke algebra.

Facts & Assumptions

Given: The isomorphism $Φ$ with $Φ ([B_{i}]) = H_{i} = v (T_{i} + 1)$ and $Φ (v X) = v Φ (X)$ , the Euler class $χ$ of Euler classes of Rouquier complexes are homotopy invariant and multiplicative, and the generator complexes of The positive and negative Rouquier generator complexes.

[F1]

Classes of the generators. $χ (F_{i}) = [B_{i}] - [R (1)]$ and $χ (F_{i}^{- 1}) = [B_{i}] - [R (- 1)]$ in $K_{0}^{s p l i t} ({S B i m}_{n})$ ; the unit class is $[R] = 1$ , and $[R (r)] = v^{r}$ under the rule $v [X] = [X (1)]$ . (Euler classes of Rouquier complexes are homotopy invariant and multiplicative, Split Grothendieck rings of the type-A Soergel categories)

[F2]

The Hecke normalization. $Φ$ is an algebra isomorphism with $Φ ([B_{i}]) = H_{i} = v (T_{i} + 1)$ , $Φ (v X) = v Φ (X)$ , $q = v^{- 2}$ ; the quadratic relation of The type-A Hecke algebra in Soergel normalization gives $T_{i} (v^{2} T_{i} - 1 + v^{2}) = 1 = (v^{2} T_{i} - 1 + v^{2}) T_{i}$ , hence $T_{i}^{- 1} = v^{2} T_{i} - 1 + v^{2}$ (The split Grothendieck group of the Soergel category is the type-A Hecke algebra)

[F3]

Multiplicativity and invariance. $χ (C \otimes_{R} D) = χ (C) χ (D)$ for signed totalizations, $χ$ is a homotopy invariant and is multiplicative over finite tensor products; the word complex of a signed word is the corresponding iterated tensor product. (Euler classes of Rouquier complexes are homotopy invariant and multiplicative, The Rouquier complex of a braid word)

[F4]

Word independence. Homotopy equivalent word complexes for the same braid have equal Euler classes, and the sign $e (σ)$ depends only on the braid; the assignment $σ_{i} \mapsto T_{i}$ extends to the group homomorphism $B_{n} \to H_{S_{n}}^{\times}$ because the Hecke algebra is presented by the braid relations and by the invertibility of $T_{i}$ . (The Rouquier complex is well defined up to canonical homotopy equivalence, The braid group by Artin presentation, The type-A Hecke algebra in Soergel normalization)

Proof

technique · direct

1.1F1F2

By [F1] and [F2] we compute $Φ (χ (F_{i})) = Φ ([B_{i}]) - Φ ([R (1)]) = H_{i} - v = v (T_{i} + 1) - v = v T_{i}$ , and likewise $Φ (χ (F_{i}^{- 1})) = H_{i} - v^{- 1}$ ; using $T_{i}^{- 1} = v^{2} T_{i} - 1 + v^{2}$ gives $v^{- 1} T_{i}^{- 1} = v T_{i} - v^{- 1} + v = H_{i} - v^{- 1}$ , so $Φ (χ (F_{i}^{- 1})) = v^{- 1} T_{i}^{- 1}$ .

2.1F1F2step 1.1

Multiplying: $Φ (χ (F_{i})) Φ (χ (F_{i}^{- 1})) = v T_{i} \cdot v^{- 1} T_{i}^{- 1} = 1$ ; equivalently $χ (F_{i} (- 1)) \mapsto T_{i}$ and $χ (F_{i}^{- 1} (1)) \mapsto T_{i}^{- 1}$ since the shift acts by $v^{\pm 1}$ . This proves (a).

2.2F3step 1.1

For a signed word $σ$ , [F3] gives $χ (F (σ)) = \prod_{k} χ (F_{i_{k}}^{ϵ_{k}})$ ; by step 1.1 each positive letter contributes $v T_{i}$ and each negative letter $v^{- 1} T_{i}^{- 1}$ , so $Φ (χ (F (σ))) = v^{# {+ 1}} v^{- # {- 1}} \prod_{k} T_{i_{k}}^{ϵ_{k}} = v^{e (σ)} T_{β}$ where $T_{β}$ is the image of $β$ under the standard homomorphism.

3.1F4step 2.2∎

Word independence: if $σ^{'}$ is another signed word for $β$ then $F (σ) ≃ F (σ^{'})$ by [F4], so $χ (F (σ)) = χ (F (σ^{'}))$ ; and $e (σ) = e (σ^{'})$ because inverse pairs have exponent zero and the two Artin relations have the same exponent on both sides. Hence the assignment $β \mapsto v^{- e (β)} χ (F (β))$ is well defined, and composing with $Φ$ gives the standard homomorphism $σ_{i} \mapsto T_{i}$ .

Remarks

The unit factors $v^{\pm 1}$ are exact and cannot be dropped: the design's shorthand "match $T_{i}^{\pm 1}$ " suppresses them, and the equalities displayed in (a) are the exact normalization of the library's shift convention. The proposition is a decategorification statement at the level of classes; it does not assert that the class map determines the homotopy type, and indeed the counterexample of this pair shows that it does not.

Depends on

Used by

Equal Euler classes do not by themselves prove homotopy-equivalent complexes Counterexample

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Catharina Stroppel, Categorification: tangle invariants and TQFTs, Proc. Int. Cong. Math. 2022, Vol. 2, EMS Press, pp. 1312-1353 (CC BY 4.0) (standard reference, not scraped)
Raphaël Rouquier, Categorification of the braid groups, arXiv:math/0409593v1 (30 September 2004), §3 "The 2-braid group" (standard reference, not scraped)
Eugene Gorsky, Oscar Kivinen, José Simental, Algebra and geometry of link homology: Lecture Notes from the IHES 2021 Summer School, Bull. London Math. Soc. 55 (2023) 537-591, §3.1 (standard reference, not scraped)