Inverse Kazhdan–Lusztig polynomials

For $x,w\in S_n$ set $q_{x,w}^{\mathrm{\prime }}:={\sum }_{m\ge 0}(-1{)}^m{\sum }_{x=z_0<z_1<\cdots <z_m=w}{p}_{z_0,z_1}{p}_{z_1,z_2}\cdots p_{z_{m-1},z_m}\in A,$ where the inner sum is over strictly increasing Bruhat chains from $x$ to $w$, the $m=0$ chain occurs only when $x=w$, and $p_{u,v}$ are the coefficients of the Kazhdan–Lusztig basis (Existence and uniqueness of the Kazhdan–Lusztig basis). Every chain lies in a finite Bruhat interval by Basic properties of the Bruhat order on $S_n$, so the sum is finite. If $x\le ̸w$ there are no chains; if $x=w$, the empty chain gives $q_{w,w}^{\mathrm{\prime }}=1$. If $x<w$, each chain has at least one factor $p_{u,v}\in v\mathbb{Z}[v]$ with $u<v$, so $q_{x,w}^{\mathrm{\prime }}\in v\mathbb{Z}[v]$.

Let $P=(p_{x,w})$ and $Q^{\mathrm{\prime }}=(q_{x,w}^{\mathrm{\prime }})$. The matrix $N:=P-I$ is strictly triangular on the finite Bruhat poset, hence nilpotent; the $(x,w)$ entry of $N^m$ is the sum of products over chains of $m$ strict steps. Therefore $Q^{\mathrm{\prime }}=I-N+N^2-\cdots +(-1{)}^M{N}^M=(I+N{)}^{-1}=P^{-1},$ where $M$ can be any integer at least the maximum strict-chain length. In particular, $Q^{\mathrm{\prime }}P=P{Q}^{\mathrm{\prime }}=I$ and ${\sum }_z{q}_{x,z}^{\mathrm{\prime }}{p}_{z,w}={\delta }_{x,w}={\sum }_z{p}_{x,z}{q}_{z,w}^{\mathrm{\prime }}\mathrm{.}$

The inverse Kazhdan–Lusztig polynomials in the classical sign convention are $q_{x,w}:=\mathrm{s}\mathrm{g}\mathrm{n}(x)\mathrm{s}\mathrm{g}\mathrm{n}(w)q_{x,w}^{\mathrm{\prime }}$. If $\mathrm{\Sigma }$ is the diagonal matrix with entries ${\mathrm{\Sigma }}_{x,x}=\mathrm{s}\mathrm{g}\mathrm{n}(x)$, then $Q=(q_{x,w})=\mathrm{\Sigma }{Q}^{\mathrm{\prime }}\mathrm{\Sigma }$. Equivalently, if $D_x$ is the basis of $H^{\ast }={\mathrm{H}\mathrm{o}\mathrm{m}}_A(H,A)$ dual to the Kazhdan–Lusztig basis, then $q_{x,w}^{\mathrm{\prime }}=D_x(H_w),$ since $H_w={\sum }_z{q}_{z,w}^{\mathrm{\prime }}{\underset{‾}{H}}_z$.

The finite chain inverse and dual-basis description use the locally proved unitriangular basis and coefficient clauses of Existence and uniqueness of the Kazhdan–Lusztig basis. Coefficientwise positivity is not required.