Bruhat intervals and the $R$-coefficients

Let $n\ge 1$ and $y,w\in S_n$. The Bruhat interval $[y,w]:=\{z\in S_n:y\le z\le w\}$ is finite, with Bruhat order as in Basic properties of the Bruhat order on $S_n$ and The Bruhat order on $S_n$ by rank inequalities. The rank-order definition uses the zero-based model of $S_n$, while $H_v(n)$ uses the one-based model; throughout, identify them by the order-preserving shift $i\mapsto i+1$ on inputs and values.

The $R$-coefficients $r_{y,w}\in A=\mathbb{Z}[v^{\pm 1}]$ are the unique coefficients in the standard-basis expansion $\overline{H_w}=H_{w^{-1}}^{-1}={\sum }_{y\in S_n}{r}_{y,w}{H}_y,$ where the bar is the involution from The Hecke bar involution is well defined and $\{H_y:y\in S_n\}$ is the standard basis of The normalized type-A Hecke algebra and its bar involution. The sum has finite support because $S_n$ is finite. In rank one, $\overline{H_{s_i}}=H_{s_i}^{-1}=H_{s_i}+(v-v^{-1})H_{\mathrm{i}\mathrm{d}}$, so $r_{\mathrm{i}\mathrm{d},s_i}=v-v^{-1}$.

The support, diagonal, and parity properties of these coefficients are stated and proved in The $R$-coefficient recursion, support, degree bounds and inversion. This page uses the coefficient normalization given by the displayed bar expansion.