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The Kazhdan–Lusztig inversion formula

Facts & Assumptions

Given: $n \geq 1$ and the finite standard and Kazhdan–Lusztig bases of $H_{v} (n)$ .

[F1]

Bar-duality gives $P = R \overset{ˉ}{P}$ , and $R \overset{ˉ}{R} = \overset{ˉ}{R} R = I$ (Existence and uniqueness of the Kazhdan–Lusztig basis, The $R$ -coefficient recursion, support, degree bounds and inversion).

[F2]

The chain-defined matrix $Q^{'}$ is the two-sided inverse of $P$ , and the classical sign matrix $S$ gives $Q = S Q^{'} S$ (Inverse Kazhdan–Lusztig polynomials).

[F3]

Bruhat intervals in $S_{n}$ are finite (Basic properties of the Bruhat order on $S_{n}$ ).

Statement

Let $P = (p_{x, w})$ , $Q^{'} = (q_{x, w}^{'})$ and $R = (r_{x, y})$ be the triangular matrices of Existence and uniqueness of the Kazhdan–Lusztig basis, Inverse Kazhdan–Lusztig polynomials and The $R$ -coefficient recursion, support, degree bounds and inversion. Then (a) $P = R \overset{ˉ}{P}$ , $\overset{ˉ}{P} = \overset{ˉ}{R} P$ , and $R \overset{ˉ}{R} = \overset{ˉ}{R} R = 1$ ; (b) $Q^{'} P = P Q^{'} = 1$ , and the inversion formulas $\overline{q_{x, w}^{'}} = \sum_{x \leq z \leq w} q_{x, z}^{'} r_{z, w}, \overline{q_{x, w}} = \sum_{x \leq z \leq w} q_{x, z} \overline{r_{z, w}}$ hold, i.e. ${\overset{ˉ}{Q}}^{'} = Q^{'} R$ and $\overset{ˉ}{Q} = Q \overset{ˉ}{R}$ ; (c) for the dual basis $D_{x} ({\underline{H}}_{w}) = δ_{x, w}$ from Inverse Kazhdan–Lusztig polynomials, $D_{x} (H_{w}) = q_{x, w}^{'}$ . The inversion is proved from the bar-duality relations alone (no finite case check).

Proof

technique · compare the coefficient matrices and use the chain inverse defining $Q'$

1.1F1algebra

Bar-duality matrices. Entrywise bar applied to $P = R \overset{ˉ}{P}$ gives $\overset{ˉ}{P} = \overset{ˉ}{R} P$ , because bar is an involution and is multiplicative on matrices over the commutative coefficient ring. The R theorem gives both $R \overset{ˉ}{R} = I$ and $\overset{ˉ}{R} R = I$ . Thus (a) holds.

2.1F1F2F3step 1.1algebra

The inverse matrix identity. By [F2], $Q^{'} = P^{- 1}$ . From $P = R \overset{ˉ}{P}$ , inversion gives $Q^{'} = {\overset{ˉ}{P}}^{- 1} R^{- 1} = {\overset{ˉ}{Q}}^{'} \overset{ˉ}{R}$ , since ${\overset{ˉ}{P}}^{- 1} = \overline{P^{- 1}} = {\overset{ˉ}{Q}}^{'}$ and $R^{- 1} = \overset{ˉ}{R}$ . Applying entrywise bar yields ${\overset{ˉ}{Q}}^{'} = Q^{'} R$ . Its $(x, w)$ entry is $\overline{q_{x, w}^{'}} = \sum_{z} q_{x, z}^{'} r_{z, w}$ ; triangular support restricts this finite sum to $x \leq z \leq w$ .

3.1F1F2F3step 2.1algebra

The signed inverse formula. Let $S$ be diagonal with $S_{x, x} = s g n (x)$ , so $S^{2} = I$ . From [F2], $Q = S Q^{'} S$ ; from the R bar-symmetry, $\overset{ˉ}{R} = S R S$ . Therefore $\overset{ˉ}{Q} = S {\overset{ˉ}{Q}}^{'} S = S Q^{'} R S = (S Q^{'} S) (S R S) = Q \overset{ˉ}{R}$ , whose $(x, w)$ entry is $\overline{q_{x, w}} = \sum_{z} q_{x, z} \overline{r_{z, w}}$ . Triangular support again restricts to $x \leq z \leq w$ .

4.1F1F2F3step 1.1step 2.1algebra∎

Dual-basis interpretation. Since $({\underline{H}}_{w})$ is a basis of the finite free module $H_{v} (n)$ , its coordinate functionals $D_{x}$ form the dual basis and satisfy $D_{x} ({\underline{H}}_{w}) = δ_{x, w}$ . Put $d_{x, w} : = D_{x} (H_{w})$ . Evaluating on ${\underline{H}}_{w} = \sum_{z} p_{z, w} H_{z}$ gives $D P = I$ ; because $P$ is invertible, $D = P^{- 1} = Q^{'}$ , so $D_{x} (H_{w}) = q_{x, w}^{'}$ . Conversely, if the dual evaluations are $q_{x, w}^{'}$ , the identity $Q^{'} P = I$ gives $D_{x} ({\underline{H}}_{w}) = δ_{x, w}$ . This proves (c). All sums are finite by [F3], and no choice principle is used.

Remarks

The matrix inversion uses the locally proved basis/bar-duality clauses and the finite chain inverse with its sign convention. Coefficientwise positivity is not required.

Depends on

Used by

The R- and Kazhdan–Lusztig recursions on a small singular interval Example

Dependency tree · two levels

15 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

G. Lusztig, Hecke Algebras with Unequal Parameters, revised version arXiv:math/0208154v2 — §10.1–10.2: the inverse chain formula, inverse-matrix identities, and signed convention; §10.7: the dual-basis interpretation. (standard reference, not scraped)