Kazhdan–Lusztig R-Polynomials Overview

• Kazhdan–Lusztig R-polynomials are recursively defined invariants in Coxeter groups that capture the transition from standard to Kazhdan–Lusztig bases.
• They connect combinatorial data from Bruhat graphs with geometric insights, such as detecting singularities in Schubert varieties.
• R-polynomials underpin computations in Hecke algebras and have extensions to affine, parabolic, and signed variants, impacting representation theory.

Kazhdan-Lusztig R-polynomials are fundamental combinatorial and algebraic invariants that arise in the context of Coxeter groups, Hecke algebras, and the geometry of Schubert varieties. Given a Coxeter system $(W,S)$ and elements $u, v \in W$ with $u \leq v$ in the Bruhat order, the R-polynomial $R_{u,v}(q)$ is defined recursively and governs many aspects of the representation theory of Hecke algebras, as well as encoding subtle geometric properties of singularities in Schubert varieties. These polynomials serve as the structural coefficients in the transition from the standard basis to the Kazhdan-Lusztig basis of the Hecke algebra and are intricately connected with increasing and decreasing paths in Bruhat graphs, reflection orders, and combinatorial invariants associated with Bruhat intervals.

1. Definition and Recursive Structure

The classical Kazhdan-Lusztig R-polynomials $R_{u,v}(q)$ are defined for $u, v \in W$ ($u \leq v$ in the Bruhat order) recursively by:

• $R_{u,u}(q) = 1$ for all $u$,
• if $u < v$ and $s$ is a simple reflection such that $s v < v$, then

$$R_{u,v}(q) = \begin{cases} R_{su,sv}(q) & \text{if } su > u, \ (q-1) R_{u,sv}(q) + q R_{su,sv}(q) & \text{if } su < u. \end{cases}$$

This recursion mirrors the structure of the Bruhat graph and the action of simple reflections. The R-polynomials appear as structural coefficients in the inversion formula for the Hecke algebra:

$$(T_{v^{-1}})^{-1} = \sum_{u \leq v} \varepsilon_u \varepsilon_v R_{u,v}(q) T_u,$$

where $\varepsilon_w = (-1)^{\ell(w)}$ and $T_w$ denotes the standard Hecke algebra basis element associated to $w$.

Theorem \ref{nth1} in (Kobayashi, 2012) establishes that when $R_{u,v}(q)$ is expanded as a polynomial in $(q-1)$, all its coefficients are nonnegative:

$$R_{u,v}(q) = \sum_{n = a(u,v)}^{\ell(u,v)} c_n (q-1)^n,$$

with $c_n \geq 0$ and $a(u,v)$ the “absolute length.”

2. Combinatorial and Geometric Significance

R-polynomials not only serve an algebraic role in the Hecke algebra but also encapsulate deep combinatorial and geometric data:

• Bruhat Graphs: The coefficients of $(q - 1)^k$ in $R_{u,v}(q)$ can be interpreted as counting certain paths or substructures in the Bruhat graph of $W$ (Kobayashi, 2012).
• Rational Singularities: The quadratic $(q-1)^2$ coefficient of the sum $\sum_{x \leq v \leq w} R_{x,v}(q)$ detects the presence of singularities in the Bruhat interval $[u, w]$. Explicitly, an excess over the binomial coefficient $\binom{\ell(x,w)}{2}$ indicates singularity.

Table: Structure and Properties of R-polynomials

Property	Expression/Formulation	Reference
Recursion	As described above	(Kobayashi, 2012)
Nonnegativity in $(q-1)$-basis	Coeffs of $(q-1)^n$ are $\geq 0$	Theorem 1, (Kobayashi, 2012)
Rational smoothness criterion	Quadratic coefficient matches $\binom{\ell}{2}$ iff smooth	Theorem 2, (Kobayashi, 2012)
Path model (combinatorial)	Sum over increasing paths in Bruhat order	(Chen et al., 2013)

The strict inequality in Kazhdan-Lusztig polynomials (Theorem 3, (Kobayashi, 2012)) states that for a singular element $u$ in $[u, w]$, there exists a cover $v$ so that $P_{u,w}(1) > P_{v,w}(1) > 0$, supporting a descent property along edges of the Bruhat graph—this refines the understanding of singularities at the combinatorial level.

3. Inversion and Reflection Principles

Kazhdan-Lusztig R-polynomials satisfy a powerful inversion formula (Chen et al., 2013):

$$\sum_{w \in [u, v]} (-1)^{\ell(w)-\ell(u)} R_{u,w}(q) R_{w,v}(q) = \delta_{u,v}.$$

A combinatorial proof is established via the notion of “V-paths,” which comprise a decreasing path from $u$ to $w$ and an increasing path from $w$ to $v$. The “reflection principle” (an involution on the set of V-paths) exchanges certain pairs of paths and cancels their contributions, yielding the desired delta function. This principle not only proves the inversion formula but also gives insight into symmetric properties of the Bruhat order, such as the equidistribution of even- and odd-length elements in Bruhat intervals.

For the symmetric group $S_n$, the inversion formula admits further refinements: restricting to the set of permutations with a prescribed value at a fixed position yields explicit evaluations of sums of products of R-polynomials (see formulas (5)–(7) in (Chen et al., 2013)).

4. Connection to Kazhdan-Lusztig and Signed Polynomials

R-polynomials are auxiliary to the Kazhdan-Lusztig polynomials $P_{u, v}(q)$:

$$q^{\ell(u,v)} P_{u,v}(q^{-1}) = \sum_{u \leq x \leq v} R_{u, x}(q) P_{x, v}(q).$$

This coupling is central to the computation of intersection cohomology of Schubert varieties and the decomposition of Verma modules (Yee, 2012).

Signed Kazhdan-Lusztig polynomials incorporate signature data of invariant Hermitian forms. The key relationship, established in (Yee, 2012), is:

$$P^{(\lambda, w_0)}_{x, y}(q) = (-1)^{\epsilon(x\lambda - y\lambda)} \cdot P_{x, y}(-q),$$

where the sign is determined by a $\mathbb{Z}_2$-grading that measures the parity of the number of noncompact roots in the weight difference.

For R-polynomials, the implication is that signature information can be obtained via classical R-polynomials by a change of variable $q \to -q$ together with a sign twist, enabling streamlined computations of signature characters in representation theory.

5. Extensions and Generalizations

Recent advancements have enriched the R-polynomial framework beyond finite Coxeter groups:

• Parabolic Kazhdan-Lusztig R-polynomials: Parabolic analogues are defined for pairs in minimal coset representative sets with respect to parabolic subgroups, with explicit combinatorial formulas available in cases such as $J = S \setminus \{s_i\}$ (Fan et al., 2015).
• Periodic and Affine R-polynomials: For extended affine or Kac-Moody groups, R-polynomials are described using paths in a “doubled” Bruhat graph or using the path model in apartments of masures (Watanabe et al., 2016, Hébert et al., 7 Oct 2024). The formulas generalize to count intersections in affine flag varieties and are integral to defining double-affine Kazhdan–Lusztig polynomials (Muthiah, 2019).
• Signed and Weighted Variants: The manipulations $q \mapsto -q$ and sign gradings extend the classical R-polynomial framework to account for Hermitian structure in Harish-Chandra and highest weight modules, offering a more direct route to analyze unitary duals and signature phenomena (Yee, 2012).

6. Implications and Future Directions

The structure and properties of R-polynomials have a multitude of implications:

• Signature Character Computation: The relationship between signed and classical polynomials enables explicit computation of signatures of invariant Hermitian forms on highest weight modules, contributing to the characterization of unitary duals for real reductive Lie groups (Yee, 2012).
• Singularity Detection: The nonnegativity and quadratic criteria link combinatorial data on Bruhat intervals directly to geometric singularities of Schubert varieties. In particular, inspection of $(q-1)^2$ coefficients gives a direct combinatorial test for rational smoothness (Kobayashi, 2012).
• Categorification and Geometric Representation Theory: The combinatorial properties of R-polynomials underpin categorical structures in the Hecke category (e.g., Soergel bimodules), play a role in diagrammatic and categorical proofs (light leaves bases, (Plaza, 2018)), and interface with modern geometric representation theory.
• Further Research Directions: Generalization to the non-equal rank case, analysis of functorial properties under Zuckerman and Bernstein functors, and the exploration of categorical or geometric interpretations of the sign twist relating classical and signed polynomials remain open for in-depth exploration (Yee, 2012).

7. Summary Table of Key Relationships

Object	Related Polynomial	Transformation/Role
Classical KL poly	$P_{u,v}(q)$	Encodes multiplicity, intersection cohomology
Signed KL poly	$P^{(\lambda, w_{0})}_{x, y}(q)$	$(-1)^{\epsilon(.)} P_{x, y}(-q)$ (signature twist)
R-polynomial	$R_{u,v}(q)$	Structure coefficients; detects paths in Bruhat graph
Signed R-poly	$(-1)^{\epsilon(.)} R_{u,v}(-q)$	Captures signature data from classical R-polynomial
Parabolic R-poly	$R_{u,v}^{J,x}(q)$	Recursive structure, explicit combinatorial statistics (Fan et al., 2015)

Understanding the intricate interplay between these polynomials, their combinatorial bases, and geometric interpretations is central to ongoing advances in the algebraic and geometric paper of Coxeter group symmetries, Hecke algebras, and representation theory.