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Positivity of Kazhdan-Lusztig polynomials, mu-coefficients and structure constants

Facts & Assumptions

Given: The Kazhdan-Lusztig basis ${\underline{H}}_{w}$ of the type- $A$ Hecke algebra with the standard basis $H_{w}$ and the transition coefficients $h_{y, w} = p_{y, w}$ of the expansion ${\underline{H}}_{w} = \sum_{y \leq w} h_{y, w} H_{y}$ (The type-A Kazhdan–Lusztig basis has elementary triangular bar properties, Kazhdan–Lusztig polynomials in the classical $q$ -normalization); the standard pairing $⟨ H_{x}, H_{y} ⟩ = δ_{x y}$ ; the character $ch$ with its $Δ$ -multiplicity definition and multiplicativity (Standard graph bimodules, support filtrations and characters, The type-A standard character is multiplicative); and the inverse Kazhdan-Lusztig coefficients defined by $H_{w} = \sum_{y \leq w} g_{y, w} {\underline{H}}_{y}$ (Inverse Kazhdan–Lusztig polynomials).

[F1]

Soergel's character theorem: $ch (B_{w}^{R}) = {\underline{H}}_{w}$ for every $w \in S_{n}$ (Soergel's character conjecture for type A).

[F2]

The character of a Soergel bimodule is a nonnegative integer combination in the standard basis: $ch (M) = \sum_{y, d} (M : Δ_{y} (d)) v^{d} H_{y}$ with multiplicities $(M : Δ_{y} (d)) \in Z_{\geq 0}$ ; the character is additive and multiplicative and satisfies $ch (M (d)) = v^{d} ch (M)$ (Standard graph bimodules, support filtrations and characters, The type-A standard character is multiplicative).

[F3]

The class map $Φ$ sends $[B_{w}^{R}]$ to ${\underline{H}}_{w}$ and is a ring isomorphism from the split Grothendieck ring onto the Hecke algebra; the split Grothendieck ring is free with basis the classes of the normalized indecomposables, so a decomposition $B_{x}^{R} B_{y}^{R} = ⨁_{z} B_{z}^{R} (m_{z})$ with graded multiplicities $m_{z} \in Z_{\geq 0} [v^{\pm 1}]$ yields the Hecke identity ${\underline{H}}_{x} {\underline{H}}_{y} = \sum_{z} m_{z} {\underline{H}}_{z}$ (The split Grothendieck group of the Soergel category is the type-A Hecke algebra, Indecomposable labels, characters and filtration ranks agree after real scalar extension).

Statement

Write ${\underline{H}}_{w} = \sum_{y \leq w} h_{y, w} H_{y}$ for the expansion of the Kazhdan-Lusztig basis element in the standard basis and ${\underline{H}}_{x} {\underline{H}}_{y} = \sum_{z} μ_{x, y}^{z} {\underline{H}}_{z}$ for the structure constants of the Kazhdan-Lusztig basis. Then:

(1) $h_{y, w} \in Z_{\geq 0} [v]$ for all $y \leq w$ ; equivalently the Kazhdan-Lusztig polynomials $P_{y, w} (q) = v^{- (ℓ (w) - ℓ (y))} h_{y, w} (v)$ with $q = v^{- 2}$ have nonnegative integer coefficients, and in particular the $μ$ -coefficients $μ (y, w) = [v] h_{y, w}$ are nonnegative;

(2) $μ_{x, y}^{z} \in Z_{\geq 0} [v^{\pm 1}]$ : the structure constants are nonnegative, and $μ_{x, y}^{z}$ is the character of the multiplicity space of $B_{z}^{R}$ in the tensor product $B_{x}^{R} \otimes_{S} B_{y}^{R}$ , i.e. its graded multiplicity;

(3) the expansion of the product $ch (B_{x}) ch (B_{y})$ in the standard basis has nonnegative coefficients, namely the $Δ$ -multiplicities of $B_{x} B_{y}$ .

No positivity of Kazhdan-Lusztig polynomials is assumed anywhere; the corollary is the local proof of the positivity clause from the character theorem.

Proof

technique · direct; read the multiplicities of the $\Delta$-filtration through the character theorem and compare coefficients in the two bases

1.1givenF1F2

The coefficients in the standard basis (1). By [F1] we have $ch (B_{w}^{R}) = {\underline{H}}_{w}$ ; expanding both sides in the standard basis, the left side has coefficients given by the $Δ$ -multiplicities, which are nonnegative integers by [F2], and the right side has coefficients $h_{y, w}$ by definition. Hence $h_{y, w} = \sum_{d} (B_{w} : Δ_{y} (d)) v^{d} \in Z_{\geq 0} [v]$ . Passing to the classical normalization $P_{y, w} (v^{- 2}) = v^{- (ℓ (w) - ℓ (y))} h_{y, w}$ of the Kazhdan-Lusztig polynomial displays its coefficients as the same nonnegative multiplicities and therefore nonnegative integers; in particular the $μ$ -coefficient, the coefficient of $v$ in $h_{y, w}$ , is a sum of multiplicities and hence nonnegative. This proves (1).

2.1step 1.1F3

The structure constants (2). The tensor product decomposes into finitely many normalized indecomposables with graded multiplicities, $B_{x}^{R} B_{y}^{R} = ⨁_{z} B_{z}^{R} (m_{z})$ with $m_{z} \in Z_{\geq 0} [v^{\pm 1}]$ . Applying the class map $Φ$ of [F3] and using multiplicativity, ${\underline{H}}_{x} {\underline{H}}_{y} = Φ ([B_{x}^{R} B_{y}^{R}]) = \sum_{z} m_{z} {\underline{H}}_{z}$ , so that $μ_{x, y}^{z} = m_{z}$ ; the multiplicity space of $B_{z}^{R}$ in the tensor product has character $m_{z}$ by definition of the multiplicity, which proves (2).

3.1step 1.1step 2.1F2∎

The standard-basis product (3). By [F2] the character of the tensor product $B_{x} B_{y}$ is a nonnegative integer combination of standard basis elements, $ch (B_{x} B_{y}) = \sum_{y^{'}, d} (B_{x} B_{y} : Δ_{y^{'}} (d)) v^{d} H_{y^{'}}$ , and by multiplicativity of the character and [F1] it equals the product $ch (B_{x}) ch (B_{y}) = {\underline{H}}_{x} {\underline{H}}_{y}$ . Comparing coefficients gives that the expansion of the product in the standard basis has exactly the $Δ$ -multiplicities as coefficients, and these are nonnegative integers. This proves (3) and completes the corollary.

Remarks

The positivity of the Kazhdan-Lusztig polynomials is thereby a corollary of the simultaneous induction of this page, not an input: the only ingredients are the character theorem, the $Δ$ -multiplicity definition of the character and the multiplicativity of the class map.
The three clauses are the same computation read in the two bases of the Hecke algebra; the graded multiplicity of an indecomposable in a tensor product is intrinsically well defined by the Krull-Schmidt decomposition and is a polynomial with nonnegative coefficients.

Depends on

Used by

Existence and uniqueness of the Kazhdan–Lusztig basis Theorem

Dependency tree · two levels

42 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, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136) (standard reference, not scraped)
Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525) (standard reference, not scraped)
Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017) (standard reference, not scraped)