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The diagrammatic character is the split $K_{0}$ Hecke isomorphism

Statement

Let $n \geq 2$ , let $D$ be the type-A diagrammatic Soergel category over $k = Q$ and $Kar (D)$ its Karoubi envelope (The type-A diagrammatic Soergel category and its candidate bimodule functor), with the indecomposables $D_{w}$ of Indecomposable type-A diagrammatic Soergel objects are indexed by permutations and shifts. For an object $B$ of $Kar (D)$ put $c h (B) : = \sum_{w \in S_{n}} grk {Hom}_{Kar (D_{\geq w})} (B, D_{w}) {\tilde{T}}_{w} \in H_{S_{n}},$ where $grk$ is the graded rank of a free graded $R$ -module, $D_{\geq w}$ is the quotient by the ideal of morphisms factoring through Bruhat cells $y \geq̸ w$ (so maps into a reduced-word object for $w$ are taken modulo lower terms), and $H_{\underline{w}} = H_{i_{1}} \dots H_{i_{r}}$ is the product of the normalized generators along a reduced word $\underline{w}$ for $w$ (the underlined symbol is reserved for reduced-word products, while the standard basis elements are the ${\tilde{T}}_{w} = v^{ℓ (w)} T_{w}$ ) (The standard basis of the type-A Hecke algebra and its multiplication rule, The type-A Hecke algebra in Soergel normalization). Then $c h$ descends to the split Grothendieck ring $K_{0}^{s p l i t} (Kar (D))$ of Split Grothendieck rings of the type-A Soergel categories, it is a homomorphism of $Z [v, v^{- 1}]$ -algebras $c h : K_{0}^{s p l i t} (Kar (D)) ⟶ H_{S_{n}},$ it satisfies $c h ([D_{i}]) = H_{i} = v (T_{i} + 1)$ for every simple reflection and $c h (v X) = v c h (X)$ , and it is an isomorphism: the classes $[D_{w}]$ are a $Z [v, v^{- 1}]$ -basis of $K_{0}^{s p l i t} (Kar (D))$ and their images $c h (D_{w}) = {\tilde{T}}_{w} + \sum_{y < w} g_{y, w} {\tilde{T}}_{y}$ are triangular with unit diagonal in the standard basis ${{\tilde{T}}_{w}}$ of The standard basis of the type-A Hecke algebra and its multiplication rule. For $n \leq 1$ both sides are $Z [v, v^{- 1}]$ and $c h$ is the identity.

Facts & Assumptions

Given: The diagrammatic category $D$ over $k = Q$ with its presentation, the Karoubi envelope $Kar (D)$ , the split Grothendieck ring of Split Grothendieck rings of the type-A Soergel categories, and the Hecke algebra $H_{n}$ of The type-A Hecke algebra in Soergel normalization.

[F1]

$K_{0}^{s p l i t} (Kar (D))$ is the free abelian group on the objects of a chosen small skeleton modulo the relations $[X] = [X^{'}] + [X^{''}]$ for $X ≅ X^{'} \oplus X^{''}$ , with product $[X] [Y] = [X \otimes Y]$ , unit $[1]$ and $v [X] = [X (1)] = [X {- 1}]$ , making it a $Z [v, v^{- 1}]$ -algebra; for an object of the skeleton the class depends only on its isomorphism class (Split Grothendieck rings of the type-A Soergel categories).

[F2]

$Kar (D)$ is Krull–Schmidt, and the objects $D_{w}$ for $w \in S_{n}$ are the indecomposables up to isomorphism and shift; every object is a finite direct sum of shifts $D_{w} (d)$ , the pair $(w, d)$ being determined by the isomorphism class and the summands being unique up to isomorphism and order (Indecomposable type-A diagrammatic Soergel objects are indexed by permutations and shifts).

[F3]

Imported from Elias–Williamson, equation (6.3), Definition 6.23, Theorem 6.25 and Corollary 6.26: for an expression $\underline{x}$ and a reduced word $\underline{w}$ for $w$ , the free module ${Hom}_{D_{\geq w}} (B_{\underline{x}}, B_{\underline{w}})$ has a light-leaf basis indexed by subexpressions of $\underline{x}$ expressing $w$ , and its graded ranks give $H_{\underline{x}} = \sum_{w} grk {Hom}_{D_{\geq w}} (B_{\underline{x}}, B_{\underline{w}}) {\tilde{T}}_{w} .$ Here $H_{\underline{x}} = H_{x_{1}} \dots H_{x_{r}}$ , while ${\tilde{T}}_{w}$ is the standard basis element. The distinguished indecomposable $D_{w}$ is the unique summand of $B_{\underline{w}}$ surviving at $w$ ; other summands have strictly lower support. The classes $[D_{w}]$ form an $A$ -basis and $c h (D_{w}) = {\tilde{T}}_{w} + \sum_{y < w} g_{y, w} {\tilde{T}}_{y}$ . Over $k = Q$ , graded direct summands of free Hom modules are free, and the character on the Karoubi envelope is an $A$ -module map. Equation (6.4) proves multiplicativity on Bott–Samelson classes, these classes span $K_{0}^{s p l i t} (Kar (D))$ , and Corollary 6.26 concludes that $c h$ is an $A$ -algebra isomorphism with $c h (B_{s}) = H_{s}$ (The type-A diagrammatic Soergel category and its candidate bimodule functor).

[F4]

Light leaves are homogeneous of degree equal to the defect $d (e) = # U_{0} - # D_{0}$ of the subexpression. Their images give the bases of the top-layer quotient Hom modules in [F3]; pasted double leaves give bases of the full Hom spaces between Bott–Samelson objects (The type-A diagrammatic Soergel category and its candidate bimodule functor, Double leaves form graded $R$ -bases of type-A diagrammatic Hom spaces).

[F5]

$H_{n}$ is the $A$ -algebra with $A = Z [v, v^{- 1}]$ presented by the $T_{i}$ with the quadratic, braid and commutation relations, $H_{i} = v (T_{i} + 1)$ and $T_{i} = v^{- 1} H_{i} - 1$ ; the family ${H_{\underline{w}}}$ , $H_{\underline{w}}$ the product of the $H_{i}$ along a reduced word $\underline{w}$ for $w$ (one reduced word chosen per element), is an $A$ -basis of $H_{n}$ with $H_{\underline{w}} = {\tilde{T}}_{w} + \sum_{ℓ (u) < ℓ (w)} a_{u} {\tilde{T}}_{u}$ (The type-A Hecke algebra in Soergel normalization, The standard basis of the type-A Hecke algebra and its multiplication rule).

Proof

1.1

The two presentations of the split group agree: [F1] presents $K_{0}^{s p l i t} (Kar (D))$ on a small skeleton modulo the relations coming from direct-sum decompositions, and by [F2] each such relation compares two decompositions of one object into indecomposables $D_{w} (d)$ , whose multisets agree; hence the class map identifies this group, together with its shift and its product, with the split group $[Kar (D)]$ of [F3] on which the character is defined, and the classes $[D_{w}]$ for $w \in S_{n}$ form a $Z [v, v^{- 1}]$ -basis while the classes $[B_{\underline{x}}]$ for words span.

F1F2F3

1.2

The character is additive and compatible with the shift: by [F3] $c h$ is additive in the argument because graded rank is additive on direct sums, and $c h (v [B]) = c h (B (1)) = v c h (B)$ ; by [F1] these are exactly the relations $[X] = [X^{'}] + [X^{''}]$ and $v [X] = [X (1)]$ of the library's presentation, so $c h$ descends to a well-defined homomorphism of $Z [v, v^{- 1}]$ -modules on $K_{0}^{s p l i t} (Kar (D))$ .

F1F3step 1.1

1.3

Subexpression multiplicities: since the module ${Hom}_{D_{\geq w}} (B_{\underline{x}}, B_{\underline{w}})$ has the light-leaf basis ${{L L}_{\underline{x}, e}}$ with $\deg {L L}_{\underline{x}, e} = d (e)$ by [F3] and [F4], its graded rank is $\sum_{e} v^{d (e)}$ over the subexpressions $e$ of $\underline{x}$ expressing $w$ ; this is the double-leaves subexpression form of the character, and it shows in particular that each coefficient of $c h (B_{\underline{x}})$ is a Laurent polynomial with non-negative integer coefficients.

F3F4

1.4

Triangularity: by [F2], $D_{w}$ is the distinguished indecomposable direct summand of a reduced-word object $B_{\underline{w}}$ . It need not be the whole object: the rank-two decomposition gives $B_{s t s} ≅ D_{s t s} \oplus B_{s}$ . The imported statement [F3] gives $c h (D_{w}) = {\tilde{T}}_{w} + \sum_{y < w} g_{y, w} {\tilde{T}}_{y}$ . Thus the matrix of ${c h (D_{w})}_{w}$ is triangular with unit diagonal over ${{\tilde{T}}_{w}}$ , hence invertible over $A$ . Equivalently its diagonal over ${T_{w}}$ is the unit $v^{ℓ (w)}$ ; this is not a unit diagonal over that unnormalized basis.

F2F3F5

2.1

The generators: for a simple reflection $s$ the imported identity $c h (B_{s}) = H_{s} = v (T_{s} + 1) = H_{i}$ of [F3] and the identification of classes of step 1.1 give $c h ([D_{i}]) = H_{i}$ ; in particular the image of $c h$ contains the algebra generators $H_{i}$ of $H_{n}$ by [F5].

F3F5step 1.2

2.2

Multiplicativity: by [F1] the product on $K_{0}^{s p l i t} (Kar (D))$ is $Z [v, v^{- 1}]$ -bilinear and by step 1.2 the character is $Z [v, v^{- 1}]$ -linear; the classes $[B_{\underline{x}}]$ of Bott–Samelson objects span by step 1.1, and for words $\underline{x}, \underline{y}$ the imported identity of [F3] gives $c h ([B_{\underline{x}}] [B_{\underline{y}}]) = c h (B_{\underline{x}} B_{\underline{y}}) = c h (B_{\underline{x}}) c h (B_{\underline{y}})$ ; expanding arbitrary elements in the spanning classes therefore gives $c h (X Y) = c h (X) c h (Y)$ for all $X, Y \in K_{0}^{s p l i t} (Kar (D))$ .

F1F3step 1.1step 1.2

3.1

Conclusion: the classes $[D_{w}]$ are a $Z [v, v^{- 1}]$ -basis of $K_{0}^{s p l i t} (Kar (D))$ by step 1.1 and their images under $c h$ are a basis of $H_{n}$ by step 1.4, so $c h$ is an isomorphism of $Z [v, v^{- 1}]$ -modules; by step 2.2 it is a homomorphism of algebras and hence an isomorphism of $Z [v, v^{- 1}]$ -algebras, with $c h ([D_{i}]) = H_{i}$ by step 2.1 and the triangular basis ${c h (D_{w})}$ as displayed. For $n \leq 1$ there are no colours; the word category has the empty word, and its additive graded Karoubi closure has finite sums and shifts of the unit. Its split Grothendieck ring and the Hecke algebra are both $A$ , and the character sends the unit to $1$ . ∎

F2F3F5step 1.1step 2.1step 2.2step 1.4

Remark

(a) What is imported and what is checked here. The mathematical content of the character isomorphism is Elias–Williamson's Corollary 6.26, recorded verbatim as imported result 7 of The type-A diagrammatic Soergel category and its candidate bimodule functor. The work of this item is the comparison of the two presentations of the split group, the identification of the coefficients with double-leaf subexpression multiplicities in the library's degree conventions, and the transport of triangularity to the library's normalized standard basis ${{\tilde{T}}_{w}}$ ; nothing beyond the imported statements and the two local Hecke items is used.

(b) The bimodule counterpart. The corresponding statement on the bimodule side, that the standard $Δ$ - and $\nabla$ -multiplicity characters are multiplicative, is proved later on this page from the split- $K_{0}$ isomorphism of the bimodule category; the simple tensoring recursion $h_{Δ} (B_{i} \otimes_{R} -) = H_{i} h_{Δ}$ of The type-A character recursion under simple Soergel tensoring is the local input there.

(c) Choice. No choice principle is used: the skeletons of [F1] are fixed sets of concrete objects, graded ranks are computed from the light-leaf bases supplied by the sources, and the triangularity argument is finite dimensional in each degree.

Depends on

Used by

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

Dependency tree · two levels

24 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, §6.5, Definition 6.23 with (6.3)–(6.4), Corollaries 6.26–6.27, PDF pp. 65–68 (standard reference, not scraped)
Libedinsky, Sur la catégorie des bimodules de Soergel, §§3–5 (standard reference, not scraped)

The diagrammatic character is the split K0K0​ Hecke isomorphism