Kazhdan–Lusztig Basis in Hecke Algebras

• Kazhdan–Lusztig basis is a specially constructed, bar-invariant set for Iwahori–Hecke algebras, defined via triangularity relative to the Bruhat order.
• It underpins the study of cell structures and induces canonical bases in modules, effectively linking representation theory, combinatorics, and geometry.
• Advanced recursive algorithms enable efficient computation of its elements, enhancing research in high-rank and affine types.

The Kazhdan–Lusztig basis is a distinguished, highly structured basis for (Iwahori–)Hecke algebras associated to Coxeter groups, with broad applications in representation theory, geometry, combinatorics, and algebraic topology. It is uniquely characterized by bar-invariance and a triangularity condition relative to the standard basis. The combinatorial, geometric, and categorical properties of the Kazhdan–Lusztig basis deeply connect it to the structure of cells, canonical bases of quantum groups, the geometry of Schubert varieties, and canonical filtrations in module categories.

1. Definition and Construction

Let $(W,S)$ be a Coxeter system with length function $\ell:W\to\mathbb{N}$. The (ungraded) Iwahori–Hecke algebra $\mathcal{H}_q(W)$ over $\mathbb{Z}[q,q^{-1}]$ has standard basis $\{T_w\mid w\in W\}$ and multiplication

$$T_s T_w = \begin{cases} T_{sw}, & \ell(sw) = \ell(w) + 1, \ (q-q^{-1})T_w + T_{sw}, & \ell(sw) = \ell(w) - 1, \end{cases}$$

with quadratic relations $(T_s-q)(T_s+q^{-1})=0$ for $s\in S$. The bar-involution sends $q\mapsto q^{-1}$ and $T_w\mapsto T_{w^{-1}}^{-1}$.

The Kazhdan–Lusztig basis $\ell:W\to\mathbb{N}$0 (in its standard normalization) is the unique $\ell:W\to\mathbb{N}$1-basis characterized by:

• $\ell:W\to\mathbb{N}$2;
• $\ell:W\to\mathbb{N}$3 (triangularity with respect to the Bruhat order).

Expressing $\ell:W\to\mathbb{N}$4 in the standard basis,

$\ell:W\to\mathbb{N}$5

where $\ell:W\to\mathbb{N}$6 are the Kazhdan–Lusztig polynomials, with $\ell:W\to\mathbb{N}$7, $\ell:W\to\mathbb{N}$8 for $\ell:W\to\mathbb{N}$9 (Yin, 2010).

For Hecke algebras with unequal parameters, the construction and relation to the length function and weight function is adjusted accordingly, but the same triangularity and bar-invariance characterize the basis (Xie, 2015).

2. Fundamental Properties, Cells, and Structure Constants

The Kazhdan–Lusztig basis provides a tool for constructing important two-sided ideals and describing the cell structure in Hecke algebras:

• The preorders $\mathcal{H}_q(W)$0, $\mathcal{H}_q(W)$1, and $\mathcal{H}_q(W)$2 on $\mathcal{H}_q(W)$3 are defined via the action of the $\mathcal{H}_q(W)$4 on each other; the associated equivalence classes are the left, right, and two-sided (Kazhdan–Lusztig) cells.
• Cell modules, constructed as quotients of sums of $\mathcal{H}_q(W)$5 over lower or equal cells, control the modular and representation-theoretic structure, and for the symmetric group, these can be described explicitly in terms of Young tableaux and the Robinson–Schensted–Knuth correspondence (Gossow et al., 2021, Yin, 2010).

Structure constants for multiplication,

$\mathcal{H}_q(W)$6

are governed by the positivity and combinatorics of the Kazhdan–Lusztig polynomials. For type $\mathcal{H}_q(W)$7, explicit cancellation-free combinatorial interpretations are possible in terms of certain path-counting models when reversal (parabolic) factorizations of basis elements exist (Parisi et al., 20 May 2026).

3. Parabolic, Quasi-Parabolic, and Generalized Kazhdan–Lusztig Bases

Given a standard parabolic subgroup $\mathcal{H}_q(W)$8, Deodhar defined a parabolic Kazhdan–Lusztig basis on the induced module $\mathcal{H}_q(W)$9:

• Each $\mathbb{Z}[q,q^{-1}]$0 ($\mathbb{Z}[q,q^{-1}]$1, minimal right coset representatives) is uniquely bar-invariant and triangular with respect to the standard basis of $\mathbb{Z}[q,q^{-1}]$2.
• The relative polynomials $\mathbb{Z}[q,q^{-1}]$3 define the change of basis, generalizing the original construction to modules (and algebra quotients) indexed by parabolic data (Yin, 2010, Lenart et al., 2016).

In type $\mathbb{Z}[q,q^{-1}]$4 and for more general reflection coset modules, quasi-parabolic KL bases are constructed using compatible bar-involutions, and fundamental inversion and triangularity properties are proved. The resulting quasi-parabolic KL polynomials satisfy inversion relations analogous to those in the classical setting (Shen et al., 2021).

4. Kazhdan–Lusztig Bases in Modules: Specht and Canonical Bases

In type $\mathbb{Z}[q,q^{-1}]$5, the Kazhdan–Lusztig basis induces a canonical basis on Specht modules, indexed by standard Young tableaux (Yin, 2010). The transition matrix between the Specht (polytabloid) basis and the Kazhdan–Lusztig basis is unitriangular with integer coefficients, and explicit formulas for the transition are derived from the parabolic Kazhdan–Lusztig theory (Im, 2019). More generally, the KL basis is upper triangular with respect to any generalized Gelfand–Tsetlin basis constructed from a multiplicity-free tower of parabolic subgroups (Haidar et al., 2024).

Quantum Schur–Weyl duality provides a structural explanation for the correspondence of various canonical and projected bases in type $\mathbb{Z}[q,q^{-1}]$6; the KL basis aligns with canonical bases of $\mathbb{Z}[q,q^{-1}]$7 for $\mathbb{Z}[q,q^{-1}]$8, with deep links to crystal, global, and canonical bases (Blasiak, 2011, Shen et al., 2021).

5. Geometric and Categorical Interpretations

The geometry underlying the Kazhdan–Lusztig basis is rich. In the equivariant $\mathbb{Z}[q,q^{-1}]$9-theory and oriented cohomology of flag varieties, the KL basis elements correspond to explicit characteristic classes, often identified with motivic Chern classes of intersection cohomology sheaves for Schubert varieties (Lenart et al., 2020). In equivariant oriented cohomology—especially for theories like the 2-parameter hyperbolic case—the (parabolic) KL–Schubert basis is the closest analogue of the classical Schubert basis. For Grassmannians, these classes coincide with those coming from Zelevinsky's small resolutions, leading to universality properties crucial for generalizing to elliptic cohomology (Lenart et al., 2016, Lenart et al., 2020).

In the affine setting, notably for extended affine Hecke algebras, the KL basis captures tensor product multiplicities for the underlying reductive group. The multiplication by Weyl characters corresponds to translations on the KL basis elements, with combinatorial formulas for structure coefficients in terms of admissible subsets and Littelmann paths (Guilhot, 2016).

In the context of matroids, a canonical KL-type basis is constructed for $\{T_w\mid w\in W\}$0-deformed Möbius algebras, transferring and generalizing some positivity properties, though many classical features (cells, duality, noncommutativity) are absent (Elias et al., 2014).

6. Algorithmic and Optimization Aspects

Kazhdan–Lusztig basis elements are classically computed recursively, a process that becomes infeasible at large ranks due to storage requirements. An algorithm (Scott–Sprowl) exists for computing individual elements without computing auxiliary elements, achieving substantial memory savings and enabling previously intractable computations in high rank or affine type (Scott et al., 2013).

The Kazhdan–Lusztig basis has also been characterized as an optimal solution to a continuous quadratic maximization problem: among all bases of a Specht module unitriangular with respect to the polytabloid basis, and invariant under all $\{T_w\mid w\in W\}$1 for $\{T_w\mid w\in W\}$2, the KL basis uniquely spans the maximal invariant cone in many cases (hooks, two-column shapes, certain small $\{T_w\mid w\in W\}$3), while Young's seminormal basis uniquely minimizes the quadratic form (Goertzen et al., 20 Apr 2026).

7. Decomposition Theorems, Lusztig's Conjectures, and Generalizations

Explicit decomposition theorems exist in certain small rank and affine cases, providing exact factorization formulas for KL basis elements over two-sided cells. In affine rank 2 (types $\{T_w\mid w\in W\}$4 and $\{T_w\mid w\in W\}$5), each two-sided cell admits a partition indexed by finite distinguished involutions, and basis elements factor as $\{T_w\mid w\in W\}$6 modulo lower cells (Xie, 2015). These decompositions have direct implications for Lusztig’s conjectures (P1–P15) on cells, orthogonality, and leading coefficients.

In general, such decomposition and partition techniques, parabolic and strip cell theory, and the Hodge theory of Soergel bimodules are expected to extend the reach of the KL basis, cell theory, and their positivity and universality properties to broader settings—though substantial combinatorial and representation-theoretic challenges remain (Xie, 2015, Lenart et al., 2020).