On factorization of matrix of Kazhdan-Lusztig polynomials

1. Problem Statement

The paper addresses the positivity of coefficients in the context of Hecke algebras associated with Coxeter systems $(W, S)$.

• Background: Kazhdan and Lusztig (1979) defined polynomials $P_{y,w}(q)$ describing the change of basis from the standard basis $\{T_w\}$ to the Kazhdan-Lusztig (KL) basis $\{C_w\}$. It was proven (by Elias and Williamson in 2014 for general Coxeter groups) that these coefficients are non-negative.
• The Gap: While positivity is known for the transition between the standard and KL bases, and for multiplication by KL basis elements (via Grojnowski-Haiman), the behavior of the transition between hybrid bases (introduced by Grojnowski and Haiman) was less explored.
• Core Question: Do the coefficients of the change of basis between two hybrid bases $T C^J$ and $T C^I$ (where $I \subseteq J \subseteq S$) have non-negative coefficients? Furthermore, can the matrix of KL polynomials be factorized into a product of matrices with non-negative entries?

2. Methodology

The authors employ a combination of algebraic combinatorics and geometric representation theory.

A. Algebraic Framework

1. Hecke Algebra & Restriction Map:
   • They define a natural restriction map $|_J: H \to H_J$ (where $H_J$ is the parabolic Hecke algebra generated by a subset $J \subseteq S$). This map sends $T_w \to T_w$ if $w \in W_J$ and $0$ otherwise.
   • They analyze the expansion of restricted elements $(T_u^{-1} C_w)|_J$ in the KL basis of $H_J$.
2. Hybrid Bases:
   • Utilizing the Grojnowski-Haiman definition, they introduce the hybrid basis $T C^J = \{T C^J_w\}_{w \in W}$.
   • $T C^J_w = T_{w_J} C_{w^J}$, where $w = w_J \cdot w^J$ is the parabolic factorization ($w_J \in W_J, w^J \in W^J$).
   • This basis interpolates between the standard basis ($J=\emptyset$) and the KL basis ($J=S$).
3. Factorization Strategy:
   • The authors aim to show that the transition matrix from $T C^J$ to $T C^I$ (for $I \subseteq J$) has non-negative polynomial entries.
   • They establish a link between the coefficients of the hybrid basis expansion and the restriction coefficients defined via the map $|_J$.

B. Geometric Framework

• For Weyl groups of symmetrizable Kac-Moody algebras, the authors provide a geometric proof of positivity.
• They utilize the categorification of the Hecke algebra via the derived category of perverse sheaves (and mixed Hodge modules) on the flag variety $G/B$.
• The KL basis elements $C_w$ correspond to intersection cohomology complexes $IC_w$.
• The restriction map and multiplication by $T_u^{-1}$ are realized as functors involving hyperbolic localization (Braden's theorem).
• They prove that these functors preserve semisimplicity (mapping pure objects to pure objects), which implies the non-negativity of the coefficients in the Grothendieck group.

3. Key Contributions and Results

Theorem A (Restriction Coefficients)

For $u \in W^J$ (minimal coset representatives) and $w \in W$, the coefficients of the expansion of $(T_u^{-1} C_w)|_J$ in the KL basis of $H_J$ are polynomials in $q$ with non-negative coefficients.

• Significance: This confirms that restricting a KL basis element (twisted by a coset representative) to a parabolic subgroup yields a positive linear combination of KL basis elements in the subalgebra.

Theorem B (Positivity of Hybrid Basis Transitions)

For subsets $I \subseteq J \subseteq S$, the expansion of a hybrid basis element $T C^J_w$ in terms of the hybrid basis $T C^I_x$ has non-negative coefficients:
$$T C^J_w = \sum_{x \in W} h^{I,J}_{x,w}(q) T C^I_x, \quad \text{where } h^{I,J}_{x,w}(q) \in \mathbb{Z}_{\geq 0}[q].$$

• Significance: This generalizes previous positivity results (Springer, Lusztig, Dyer-Lehrer) to the setting of hybrid bases.

Theorem C (Factorization of KL Matrix)

The matrix of Kazhdan-Lusztig polynomials (the transition matrix from the standard basis to the KL basis) factorizes into a product of $k$ matrices, where $k = |S|$.

• Specifically, for a chain $\emptyset = J_0 \subset J_1 \subset \dots \subset J_k = S$, the KL matrix is the product of transition matrices between adjacent hybrid bases $T C^{J_i}$ and $T C^{J_{i+1}}$.
• Since each factor in this product has non-negative entries (by Theorem B), the factorization provides a structural explanation for the non-negativity of KL polynomials.

Additional Findings

• Parabolic KL Polynomials: The authors show that parabolic KL polynomials (associated with the sign representation) are a special case of these restriction coefficients.
• Explicit Computations: They compute explicit formulas for restriction coefficients in Type $A_n$ (symmetric groups) and Dihedral groups ($I_2(m)$), verifying the positivity in these specific cases.
• Geometric Proof: Section 8 provides a rigorous geometric proof of Theorem A using mixed Hodge modules and hyperbolic localization, confirming that the positivity holds for Weyl groups of symmetrizable Kac-Moody algebras.

4. Significance and Implications

1. Structural Insight: The factorization theorem (Theorem C) decomposes the complex structure of the KL matrix into simpler, positive steps. This offers a new perspective on why KL polynomials are non-negative, linking it to the geometry of flag varieties and the properties of hybrid bases.
2. Generalization: The results extend the scope of positivity theorems beyond the standard-to-KL transition, covering the entire spectrum of hybrid bases.
3. Geometric vs. Combinatorial: The paper bridges combinatorial algebra (Hecke algebras) with deep geometric tools (perverse sheaves). While the combinatorial proof relies on the Elias-Williamson result for general Coxeter groups, the geometric proof is restricted to symmetrizable Kac-Moody types but offers a more intuitive "categorical" reason for positivity.
4. Future Directions: The authors note that while the geometric proof is limited to specific groups, the combinatorial positivity for general Coxeter groups remains an open problem for the hybrid basis transition. They suggest using Soergel bimodules (from Elias-Williamson theory) to tackle the general case in future work.

Conclusion

This paper establishes a fundamental factorization of the Kazhdan-Lusztig matrix into a product of positive matrices via hybrid bases. By defining restriction coefficients and proving their positivity both algebraically (via induction and properties of the restriction map) and geometrically (via perverse sheaves), the authors provide a robust framework for understanding the non-negativity of KL polynomials and their generalizations.