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The split Grothendieck group of the Soergel category is the type-A Hecke algebra

Statement

Let $n \geq 2$ and $k = Q$ , let $K_{0}^{s p l i t} ({S B i m}_{n})$ be the split Grothendieck ring of the type-A Soergel category (Split Grothendieck rings of the type-A Soergel categories, The type-A Soergel category ${S B i m}_{n}$ ) and let $H_{S_{n}}$ be the type-A Hecke algebra over $A = Z [v, v^{- 1}]$ with $q = v^{- 2}$ and normalized generators $H_{i} = v (T_{i} + 1)$ (The type-A Hecke algebra in Soergel normalization). Then there is an isomorphism of $Z [v, v^{- 1}]$ -algebras $Φ : K_{0}^{s p l i t} ({S B i m}_{n}) ⟶ H_{S_{n}}, Φ ([B_{i}]) = H_{i} = v (T_{i} + 1) (1 \leq i \leq n - 1), Φ (v X) = v Φ (X),$ equivalently $Φ^{- 1} (T_{i}) = v^{- 1} [B_{i}] - 1$ for every simple reflection. In particular the classes satisfy $[B_{i}]^{2} = (v + v^{- 1}) [B_{i}] in K_{0}^{s p l i t} ({S B i m}_{n}),$ in agreement with the quadratic relation $H_{i}^{2} = (v + v^{- 1}) H_{i}$ of the Hecke algebra, and $Φ$ is the unique algebra isomorphism with $Φ ([B_{i}]) = H_{i}$ and $Φ (v) = v$ , because the classes of the $B_{i}$ together with $v^{\pm 1}$ generate $K_{0}^{s p l i t} ({S B i m}_{n})$ as a $Z [v, v^{- 1}]$ -algebra.

Facts & Assumptions

Given: The type-A Soergel category ${S B i m}_{n}$ with its split Grothendieck ring $K_{0}^{s p l i t} ({S B i m}_{n})$ , the diagrammatic category $D$ with $Kar (D)$ , the equivalence $Kar (F) : Kar (D) \to {S B i m}_{n}$ , the Hecke algebra $H_{S_{n}}$ with its normalized generators $H_{i}$ , and the simple reflections $s_{i}$ .

[F1]

The diagrammatic character is an isomorphism of $Z [v, v^{- 1}]$ -algebras $c h : K_{0}^{s p l i t} (Kar (D)) \to H_{S_{n}}$ with $c h ([D_{i}]) = H_{i} = v (T_{i} + 1)$ for every simple reflection, $c h (v X) = v c h (X)$ , and $c h (D_{w}) = {\tilde{T}}_{w} + \sum_{y < w} g_{y, w} {\tilde{T}}_{y}$ triangular with unit diagonal in the standard basis ${{\tilde{T}}_{w} = v^{ℓ (w)} T_{w}}$ of The standard basis of the type-A Hecke algebra and its multiplication rule (The diagrammatic character is the split $K_{0}$ Hecke isomorphism).

[F2]

The evaluation functor $Kar (F) : Kar (D) \to {S B i m}_{n}$ is an equivalence of graded monoidal categories: it is full, faithful and essentially surjective, it preserves finite direct sums, tensor products, the unit and the shifts, and on the hom spaces it is a degree-preserving bijection (The type-A diagrammatic and bimodule Soergel categories are equivalent).

[F3]

$K_{0}^{s p l i t}$ is defined on a fixed small skeleton: $K_{0}^{s p l i t} (C) = (⨁_{X \in sk (C)} Z [X]) / ⟨ [X] - [X^{'}] - [X^{''}] : X ≅ X^{'} \oplus X^{''} ⟩$ , the product is $[X] [Y] = [X \otimes Y]$ , the shift is $v [X] = [X (1)] = [X {- 1}]$ , and for ${S B i m}_{n}$ one may take for the skeleton the graded direct summands of finite direct sums of shifts of Bott–Samelson bimodules with a fixed underlying set of words (Split Grothendieck rings of the type-A Soergel categories).

[F4]

Every object of ${S B i m}_{n}$ is a pair $(M, e)$ with $M$ a finite direct sum of shifts of Bott–Samelson products and $e$ a degree-zero idempotent; the tensor product is $(M, e) \otimes_{R} (N, f) = (M \otimes_{R} N, e \otimes f)$ , and a morphism $u : (M, e) \to (N, f)$ satisfies $f u = u = u e$ (The type-A Soergel category ${S B i m}_{n}$ ).

[F5]

The rank-one square: for a simple reflection $s$ there is an isomorphism of graded bimodules $B_{s} \otimes_{R} B_{s} ≅ B_{s} (1) \oplus B_{s} (- 1)$ , i.e. $B_{s} \otimes_{R} B_{s} ≅ B_{s} {- 1} \oplus B_{s} {1}$ , with non-isomorphic summands (The rank-one Soergel bimodule square splits).

[F6]

The normalized Hecke generators satisfy $H_{i} = v (T_{i} + 1)$ , $T_{i} = v^{- 1} H_{i} - 1$ , $q = v^{- 2}$ and $H_{i}^{2} = (v + v^{- 1}) H_{i}$ ; the products of the $H_{i}$ along reduced words are a triangular $A$ -basis of $H_{n}$ with unit diagonal against the normalized standard basis ${{\tilde{T}}_{w}}$ (The type-A Hecke algebra in Soergel normalization, The standard basis of the type-A Hecke algebra and its multiplication rule).

Proof

1.1

Comparison of the skeleta: let $(M, e)$ be a representative object of ${S B i m}_{n}$ ; by [F2] the functor $Kar (F)$ is essentially surjective with a degree-preserving bijection on hom spaces, and by [F4] $M$ is a given finite direct sum of shifts of Bott–Samelson products, so $M = F (X)$ for the corresponding finite direct sum $X$ of shifts of words in the additive closure of $D$ and $e$ has a unique preimage $\tilde{e}$ because $Kar (F)$ is injective on $End (X)$ ; the assignment $(M, e) \mapsto (X, \tilde{e})$ is compatible with direct sums, tensor products, the unit, shifts and degrees, so it induces a $Z [v, v^{- 1}]$ -algebra isomorphism $Ψ : K_{0}^{s p l i t} ({S B i m}_{n}) \to K_{0}^{s p l i t} (Kar (D))$ with $Ψ ([B_{i}]) = [D_{i}]$ and $Ψ (v X) = v Ψ (X)$ , the relations of [F3] corresponding on the two sides.

F2F3F4

2.1

The character transport: by [F1] the diagrammatic character is an isomorphism of $Z [v, v^{- 1}]$ -algebras and by step 1.1 so is $Ψ$ , hence $Φ : = c h \circ Ψ$ is an isomorphism of $Z [v, v^{- 1}]$ -algebras $K_{0}^{s p l i t} ({S B i m}_{n}) \to H_{S_{n}}$ with $Φ ([B_{i}]) = c h ([D_{i}]) = H_{i}$ and $Φ (v X) = v Φ (X)$ .

F1step 1.1

3.1

The quadratic relation: by [F5] $B_{i} \otimes_{R} B_{i} ≅ B_{i} (1) \oplus B_{i} (- 1)$ , so the product and shift rules of [F3] give $[B_{i}]^{2} = [B_{i} \otimes_{R} B_{i}] = [B_{i} (1)] + [B_{i} (- 1)] = v [B_{i}] + v^{- 1} [B_{i}] = (v + v^{- 1}) [B_{i}]$ ; the corresponding Hecke identity $H_{i}^{2} = (v + v^{- 1}) H_{i}$ is the expansion recorded in [F6], and applying the algebra isomorphism $Φ$ of step 2.1 to the displayed K-theoretic identity gives exactly that Hecke identity because $Φ ([B_{i}]) = H_{i}$ and $Φ$ is $Z [v, v^{- 1}]$ -linear.

F3F5F6step 2.1

3.2

The inverse on the generators: by [F6] $T_{i} = v^{- 1} H_{i} - 1$ , so applying the inverse of the isomorphism $Φ$ of step 2.1 gives $Φ^{- 1} (T_{i}) = v^{- 1} [B_{i}] - 1$ in $K_{0}^{s p l i t} ({S B i m}_{n})$ .

F6step 2.1

4.1

Uniqueness and conclusion: generation is deduced from the isomorphism of step 2.1, not from the skeleton description. By [F6] $T_{i} = v^{- 1} H_{i} - 1$ , so the normalized generators $H_{i}$ together with $v^{\pm 1}$ generate $H_{S_{n}}$ as a $Z [v, v^{- 1}]$ -algebra, since the $T_{i}$ present $H_{S_{n}}$ . Let $S \subseteq K_{0}^{s p l i t} ({S B i m}_{n})$ be the $Z [v, v^{- 1}]$ -subalgebra generated by the classes $[B_{i}]$ and $v$ : by step 2.1 $Φ$ is an isomorphism onto $H_{S_{n}}$ and $Φ (S)$ contains $v$ and every $Φ ([B_{i}]) = H_{i}$ , so $Φ (S) = H_{S_{n}}$ and injectivity of $Φ$ gives $S = K_{0}^{s p l i t} ({S B i m}_{n})$ . Thus the classes of the $B_{i}$ together with $v^{\pm 1}$ do generate the ring, and an algebra homomorphism out of it is determined by its values on the $[B_{i}]$ and on $v$ ; hence $Φ$ is the unique algebra isomorphism with $Φ ([B_{i}]) = H_{i}$ and $Φ (v) = v$ . The triangular basis statement is transported from [F1] along $Ψ$ : $Φ (Ψ^{- 1} ([D_{w}])) = c h ([D_{w}]) = {\tilde{T}}_{w} + \sum_{y < w} g_{y, w} {\tilde{T}}_{y}$ is a triangular basis of $H_{n}$ with unit diagonal over the normalized standard basis ${{\tilde{T}}_{w}}$ , and by [F6] the products of the $H_{i}$ along reduced words are triangular over the same standard basis, so the two bases are compared by the displayed values of $Φ$ and $Φ^{- 1}$ . ∎

F1F3F6step 2.1step 3.1step 3.2

(a) What is imported and what is proved here. The algebra isomorphism on the diagrammatic side is Elias–Williamson's character isomorphism, recorded as imported result 7 of The type-A diagrammatic Soergel category and its candidate bimodule functor and proved as an item of this page; the work of this item is the transport of that isomorphism across the equivalence of The type-A diagrammatic and bimodule Soergel categories are equivalent and the check that the transported classes of the elementary bimodules are the normalized Hecke generators.

(b) Products of elementary classes are not generally Kazhdan–Lusztig basis elements. The classes $[B_{i}]$ satisfy $[B_{i}]^{2} = (v + v^{- 1}) [B_{i}]$ and the braid-type relation $[B_{i}] [B_{i + 1}] [B_{i}] + [B_{i + 1}] = [B_{i + 1}] [B_{i}] [B_{i + 1}] + [B_{i}]$ obtained from the two rank-two decompositions; each $[B_{i}]$ is a simple Kazhdan–Lusztig basis element, but their products are generally not single basis elements; the elementary classes are the images of the normalized generators, exactly as the display $Φ ([B_{i}]) = H_{i} = v (T_{i} + 1)$ records. In particular the relation $[B_{i}] [B_{j}] = [B_{j}] [B_{i}]$ holds for distant colours only, and the ring $K_{0}^{s p l i t} ({S B i m}_{n})$ is non-commutative as soon as $n \geq 3$ , since $[B_{1} B_{2}] \neq [B_{2} B_{1}]$ in the basis transported from [F1].

(c) Choice. The ring $K_{0}^{s p l i t}$ is defined on the fixed skeletons of [F3], and the comparison of step 1.1 uses the presentation of a representative object by its underlying word data, which is part of the data of the skeleton element; no choice principle beyond the fixed skeletons already recorded in the definition of [F3] is used.

Depends on

Used by

Dependency tree · two levels

25 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

Elias–Williamson, Soergel Calculus, §§3, 5–7 (standard reference, not scraped)
Libedinsky, Gentle Introduction to Soergel Bimodules I, §§2–5 (standard reference, not scraped)