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Existence and uniqueness of the Kazhdan–Lusztig basis

Facts & Assumptions

Given: $n \geq 1$ , the normalized Hecke algebra $H_{v} (n)$ , its bar involution, and the Laurent coefficients $r_{x, y}$ of the bar images in the standard basis.

[F1]

The standard elements $H_{y}$ form an $A = Z [v^{\pm 1}]$ -basis of $H_{v} (n)$ (The normalized type-A Hecke algebra and its bar involution, The standard basis of the generic type-A Hecke algebra).

[F2]

The bar is a semilinear algebra involution with $\overline{v} = v^{- 1}$ (The Hecke bar involution is well defined).

[F3]

The coefficients satisfy $r_{x, y} = 0$ unless $x \leq y$ , $r_{y, y} = 1$ , and both matrix identities $R \overset{ˉ}{R} = \overset{ˉ}{R} R = I$ ; for $x \leq y$ , their degree bounds, parity, and top coefficient are as stated in the R-coefficient theorem (The $R$ -coefficient recursion, support, degree bounds and inversion).

[F4]

The $A$ -linear anti-automorphism $♭$ commutes with bar and sends $H_{y}$ to $H_{y^{- 1}}$ (Reversal anti-involution commutes with the Hecke bar).

[F5]

Bruhat order on $S_{n}$ is a finite graded order, strict inequalities raise length, and inversion preserves the order (Basic properties of the Bruhat order on $S_{n}$ ).

[F6]

Elias–Williamson, Corollary 1.2(1), states that the coefficients $h_{y, w}$ of their triangular bar-fixed basis belong to $Z_{\geq 0} [v]$ . Their §3.2 uses $H_{s}^{2} = 1 + (v^{- 1} - v) H_{s}$ , and Remark 3.2 fixes $q = v^{- 2}$ and $h_{y, w} = v^{ℓ (w) - ℓ (y)} P_{y, w} (v^{- 2})$ . This is the single original-source positivity fact authorized for this item; its Soergel–Hodge proof is not a local prerequisite.

Statement

For each $w \in S_{n}$ there is a unique element ${\underline{H}}_{w} \in H_{v} (n)$ with (i) $\overline{{\underline{H}}_{w}} = {\underline{H}}_{w}$ and (ii) ${\underline{H}}_{w} \in H_{w} + \sum_{y < w} v Z [v] H_{y}$ (the sum over the lower Bruhat ideal of $w$ ). The elements ${{\underline{H}}_{w}}_{w \in S_{n}}$ form an $A$ -basis of $H_{v} (n)$ , and writing ${\underline{H}}_{w} = \sum_{y \leq w} p_{y, w} H_{y}$ one has $p_{w, w} = 1$ , $p_{y, w} = 0$ unless $y \leq w$ , $p_{y, w} \in v Z [v]$ for $y < w$ , the bar-duality $p_{x, w} = \sum_{y} r_{x, y} \overline{p_{y, w}}$ (matrix form $P = R \overset{ˉ}{P}$ ), and the symmetry $p_{y^{- 1}, w^{- 1}} = p_{y, w}$ . Moreover for $y < w$ , with $d = ℓ (w) - ℓ (y)$ : $p_{y, w} = v^{d} + terms of strictly smaller degree$ and $p_{y, w} \in v^{d} Z [v^{- 2}]$ ; in particular every ${\underline{H}}_{w}$ has integer nonnegative coefficients in the standard basis with $p_{y, w}$ of fixed parity $d m o d 2$ .

Proof

technique · construct the triangular bar-fixed element by descending induction in the finite Bruhat order, then use its uniqueness and coefficient comparison

1.1F2F3F5algebra

Construct the coefficients. Fix $w$ and descend on $d = ℓ (w) - ℓ (x)$ over the finite lower Bruhat ideal ${x : x \leq w}$ , starting with $p_{w, w} = 1$ . Suppose $x < w$ and $p_{y, w}$ has been constructed for every $x < y \leq w$ , satisfying $p_{y, w} = \sum_{y \leq z \leq w} r_{y, z} \overline{p_{z, w}}$ . Put $a_{x} : = \sum_{x < y \leq w} r_{x, y} \overline{p_{y, w}}$ . Then $\begin{aligned} \overline{a_{x}} & = \sum_{x < y \leq w} \overline{r_{x, y}} p_{y, w} = \sum_{x < y \leq z \leq w} \overline{r_{x, y}} r_{y, z} \overline{p_{z, w}} \\ = - \sum_{x < z \leq w} r_{x, z} \overline{p_{z, w}} = - a_{x} . \end{aligned}$ The second equality uses the induction equations; for $x < z$ , the identity $\overset{ˉ}{R} R = I$ makes the sum over $x \leq y \leq z$ zero, and its omitted diagonal term is $\overline{r_{x, x}} r_{x, z} = r_{x, z}$ . Write $a_{x} = \sum_{m \in Z} γ_{m} v^{m}$ . Anti-invariance gives $γ_{- m} = - γ_{m}$ and $γ_{0} = 0$ . Define $p_{x, w} : = \sum_{m > 0} γ_{m} v^{m}$ . Then $p_{x, w} \in v Z [v]$ and $p_{x, w} - \overline{p_{x, w}} = a_{x}$ . The induction is finite and uses no choice principle.

2.1F1F2F3F5step 1.1algebra

Bar invariance and the coefficient equations. Set ${\underline{H}}_{w} : = \sum_{y \leq w} p_{y, w} H_{y}$ . The coefficient of $H_{x}$ in $\overline{{\underline{H}}_{w}}$ is $\sum_{x \leq y \leq w} r_{x, y} \overline{p_{y, w}}$ . For $x = w$ this is $1$ ; for $x < w$ it is $\overline{p_{x, w}} + a_{x} = p_{x, w}$ by step 1.1. If $x \leq̸ w$ , no $y \leq w$ can satisfy $x \leq y$ , so support from [F3] gives coefficient zero. Thus $\overline{{\underline{H}}_{w}} = {\underline{H}}_{w}$ and $p_{x, w} = \sum_{x \leq y \leq w} r_{x, y} \overline{p_{y, w}}$ , which is the bar-duality formula.

3.1F1F2F3step 2.1algebra

Uniqueness. If two bar-invariant elements satisfy the triangular condition, their difference is a bar-fixed sum $h = \sum_{y < w} c_{y} H_{y}$ with each $c_{y} \in v Z [v]$ . If $h \neq 0$ , choose a Bruhat-maximal $x$ in its finite support. The coefficient of $H_{x}$ in $\overline{h}$ is $\overline{c_{x}}$ : no supported $y > x$ contributes, and support of $r_{x, y}$ requires $x \leq y$ . Since $h = \overline{h}$ , $c_{x} = \overline{c_{x}}$ ; but $v Z [v]$ and $v^{- 1} Z [v^{- 1}]$ intersect only in $0$ , a contradiction. Thus the element is unique. The construction also gives $p_{w, w} = 1$ and $p_{y, w} = 0$ unless $y \leq w$ .

3.2F1F5step 2.1algebra

Basis. The transition from $(H_{w})$ to $({\underline{H}}_{w})$ is unitriangular on the finite Bruhat poset: each ${\underline{H}}_{w} = H_{w} + \sum_{y < w} p_{y, w} H_{y}$ . A finite unitriangular matrix over $A$ is invertible, so $({\underline{H}}_{w})_{w \in S_{n}}$ is an $A$ -basis.

3.3F3step 1.1step 2.1algebra

Degree, leading term, and parity. Induct on $d = ℓ (w) - ℓ (x)$ , with $p_{w, w} = 1$ . For $x < w$ , the equation in step 1.1 has $a_{x} = \sum_{x < y \leq w} r_{x, y} \overline{p_{y, w}}$ . By [F3] and induction, every term has exponents congruent to $d$ modulo $2$ . The term $y = w$ is $r_{x, w}$ , whose highest term is $v^{d}$ with coefficient $1$ . For each $y < w$ , put $d_{1} = ℓ (y) - ℓ (x)$ and $d_{2} = ℓ (w) - ℓ (y) \geq 1$ , so $d_{1} + d_{2} = d$ ; the highest degree of $r_{x, y} \overline{p_{y, w}}$ is at most $d_{1} - 1 \leq d - 2$ , since $p_{y, w} \in v Z [v]$ . Therefore $a_{x}$ has highest term $v^{d}$ with coefficient $1$ and only exponents of parity $d$ . As $p_{x, w}$ is its positive-degree part by step 1.1, it follows that $p_{x, w} = v^{d} +$ terms of strictly smaller degree and $p_{x, w} \in v^{d} Z [v^{- 2}]$ . This proves the degree and parity clauses.

4.1F4F5step 3.1algebra

Inverse-index symmetry. By [F4], $♭ ({\underline{H}}_{w})$ is bar-fixed. By [F5], inversion preserves Bruhat order, so this element has the form $H_{w^{- 1}} + \sum_{y < w} p_{y, w} H_{y^{- 1}}$ with lower terms in $v Z [v]$ . Uniqueness from step 3.1 gives $♭ ({\underline{H}}_{w}) = {\underline{H}}_{w^{- 1}}$ . Comparing coefficients yields $p_{y^{- 1}, w^{- 1}} = p_{y, w}$ .

5.1F1F2F6step 2.1step 3.1step 3.3algebra∎

Normalize and apply the authorized positivity result. The identity on $v$ and on each $H_{s}$ identifies our presented algebra with the type- $A$ algebra in Elias–Williamson’s Hecke section: the quadratic and braid relations agree by [F1], and its bar agrees on $v$ and all generators by [F2]. Its standard element $H_{w}$ is the same reduced-word product as ours. Their basis has precisely the bar-invariance and triangularity established in step 2.1, so uniqueness in step 3.1 identifies it with our ${\underline{H}}_{w}$ . Comparing standard-basis coefficients gives $p_{y, w} = h_{y, w}$ . The authorized positivity conclusion in [F6] therefore gives $p_{y, w} \in Z_{\geq 0} [v]$ . For $y < w$ , step 3.3 and $p_{y, w} \in v Z [v]$ give the polynomial $P_{y, w} (q) = v^{- d} p_{y, w}$ , $q = v^{- 2}$ , exactly as in the normalization remark of [F6]; this substitution preserves individual integer coefficients. The diagonal and unsupported coefficients are respectively $1$ and $0$ . Thus coefficientwise nonnegativity and every asserted boundary case hold.

Remarks

Existence, uniqueness, basis, support, bar-duality, degrees, leading terms, parity, inverse symmetry and the normalization comparison are proved locally. Only coefficientwise positivity invokes the owner's exact original-source fallback, recorded in research/frontier-43-complex-representation-15-kl-positivity-citation-authorization.json. The triangular construction alone does not imply positivity, and no local proof of the Soergel–Hodge theorem is asserted. All local inductions are finite and use no Choice.

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