Kazhdan–Lusztig Basis in Hecke Algebras

• Kazhdan–Lusztig basis is a distinguished set of elements in Hecke algebras defined via bar involution invariance and triangular expansions with positive combinatorial interpretations.
• The combinatorial mask model encodes defect statistics through binary sequences attached to fixed reduced words, linking subexpression selection to geometric resolutions.
• LS tree constructions and geometric approaches, such as Bott–Samelson and Zelevinsky resolutions, translate mask sets into explicit expansions in the Bruhat order.

The Kazhdan–Lusztig basis is a distinguished basis in the (Iwahori-)Hecke algebra associated to a Coxeter group, introduced in the context of representation theory, geometry of Schubert varieties, and algebraic combinatorics. For cograssmannian permutations—permutations with at most one right ascent—the combinatorics and geometry underlying this basis become particularly rich. Notably, positive combinatorial formulas exist for evaluating the Kazhdan–Lusztig polynomials and basis elements in terms of "masks": binary sequences attached to a fixed reduced word that encode subexpressions and "defect" statistics. There are deep connections between this combinatorial model and geometric constructions such as the Bott–Samelson resolution of Schubert varieties, culminating in parallel mask–set models with combinatorially and geometrically meaningful interpretations (Jones et al., 2010).

1. Definition and Structural Origins

The Kazhdan–Lusztig basis $\{C_w\}$ of the Hecke algebra over a Coxeter group $(W, S)$ is characterized by two properties:

• Bar involution invariance $\overline{C_w} = C_w$, where the bar involution is the ring and Hecke algebra automorphism $q \mapsto q^{-1}$ and $T_w \mapsto T_{w^{-1}}^{-1}$.
• A triangular expansion in terms of the standard basis: $C_w = T_w + \sum_{y<w} p_{y,w}(q) T_y$, where $p_{y,w}$ are the Kazhdan–Lusztig polynomials supported at $q$-degrees $>0$ and the sum is over $y$ less than $w$ in the (strong) Bruhat order.

For cograssmannian permutations (those with at most one right ascent), these basis elements admit explicit combinatorial models interpretable both in terms of subword masks and as indexing geometric structures in certain resolutions of Schubert varieties.

2. Deodhar’s Mask Model and Defect Statistic

Deodhar introduced an approach to Kazhdan–Lusztig polynomials and basis elements using the notion of "masks" on a fixed reduced expression $w = w_1 w_2 \ldots w_{\ell(w)}$:

• A mask is a binary string $\sigma \in \{0, 1\}^{\ell(w)}$, determining which generators in the reduced word are "active".
• Each mask gives rise to a subexpression $w^\sigma$, and carries a defect statistic $d(\sigma)$ (interpreted variously as the number of additive "defects" or "+" symbols in certain encodings).
• The Kazhdan–Lusztig basis element can be written as

$$C'_w = q^{-\frac{1}{2} \ell(w)} \sum_{\sigma \in \mathcal{E}} q^{d(\sigma)} T_{w^\sigma}$$

where $\mathcal{E}$ is an admissible set of masks, determined by boundedness conditions inherent to the definition.

This mask model provides a positive combinatorial formula for the Kazhdan–Lusztig basis elements, in contrast to the original recursive approach.

3. Lascoux–Schützenberger Tree Construction

The Lascoux–Schützenberger (LS) formula gives a combinatorial model, especially effective for cograssmannian permutations, utilizing edge-labeled rooted trees derived from the heap or string diagram of a reduced word:

• The heap for $w$ is decomposed into "segments" and "regions," with the "ridgeline" encoding a sequence of matched parentheses.
• Parentheses correspond to the formation of an explicit rooted tree $T(w)$; each leaf (valley) receives a capacity determined by the heap of a grassmannian prefix $v$ in the expression $w = v w_0^J$.
• Edge labelings of $T(w)$ correspond to families of masks (constant on non-valley regions) with specific defect positions.
• Each tree $t$ yields a lower Bruhat ideal, and the set of masks $\mathcal{E}_w$ assembled over all $t \in A_w$ encodes the full Kazhdan–Lusztig basis element as a sum over masks with explicitly calculable defects.

The algorithm translates the problem of finding admissible sets of masks to the purely combinatorial matching and labeling of trees, reflecting principal lower order ideals in the Bruhat order.

4. Bott–Samelson and Zelevinsky Geometric Constructions

The geometric perspective interprets masks as indexing the $T$-fixed points in the Bott–Samelson resolution $\pi_w : Z_w \rightarrow X_w$ of the Schubert variety:

• A point in $Z_w$ is given by a tuple $[V_1, \ldots, V_{\ell(w)}]$ of vector subspaces subject to specific incidence relations reflecting the heap structure.
• The Bialynicki–Birula decomposition yields cells $C_\sigma$ indexed by masks, whose dimension is $\dim(C_\sigma) = d(\sigma) + \ell(w^\sigma)$.
• In this setting,

$$C'_w = q^{-\frac{1}{2} \ell(w)} \sum_\sigma q^{\dim(C_\sigma) - \ell(w^\sigma)} T_{w^\sigma}$$

links the combinatorics of masks directly to the geometry of the resolution via cell dimensions.

• Zelevinsky's resolution, which is particularly fine in the cograssmannian case, provides further subdivisions corresponding to chains of partitions inside rectangles ("peaks" of the heap), and the mask model lifts accordingly.

The geometric model reveals that the defect statistic of a mask coincides with the dimension of the associated cell, intertwining combinatorics and geometry.

5. Principal Lower Order Ideals and Combinatorial Translations

Principal lower order ideals in the Bruhat order serve as a bridge between mask-based formulas and the ideal basis of the Hecke algebra:

• The ideal basis is $$B'_x = q^{-\frac{1}{2} \ell(x)} \sum_{y \leq x} T_y$$.
• For cograssmannian $w$, the Kazhdan–Lusztig basis element $C'_w$ expands as

$$C'_w = q^{-\frac{1}{2} \ell(w)} \sum_{t \in A_w} q^{|t| + \frac{1}{2}\ell(x(t))} B'_{x(t)}$$

with each $t$ corresponding to an edge-labeled tree, $|t|$ the sum of its labels, and $x(t)$ the associated permutation from the mask.

• The translation of trees to mask sets thereby realizes combinatorially positive expressions for the coefficients in the ideal basis expansion, giving a positive, nonrecursive rule for $C'_w$ in terms of the principal ideals.

This viewpoint underlines the central role played by the Bruhat order and its ideals in structuring both mask combinatorics and the algebraic description of the Kazhdan–Lusztig basis.

6. Non-Uniqueness and Admissibility of Mask Sets

A key outcome is that multiple (distinct) admissible sets of masks produce valid, positive formulas for $C'_w$:

• Both the combinatorial (LS) and geometric (Bott–Samelson/Zelevinsky) constructions yield bounded admissible collections of masks whose defect enumerations recover the correct Kazhdan–Lusztig polynomial or basis element.
• These sets are not unique, indicating intrinsic flexibility in Deodhar's framework. This non-uniqueness reflects the existence of multiple natural "expansions" for $C'_w$, depending on further combinatorial or geometric choices.
• The question of selecting a canonical mask set remains open and could motivate further axiomatization.

This multiplicity demonstrates the richness of the combinatorics behind the Kazhdan–Lusztig basis for cograssmannian permutations.

7. Connections, Applications, and Theoretical Implications

The interplay between the combinatorial and geometric models for the Kazhdan–Lusztig basis has broad implications:

• The mask set constructions facilitate positive, explicit formulas and computational methods for evaluating Kazhdan–Lusztig polynomials in special cases.
• The geometric correspondence between mask defect statistics and cell dimensions clarifies deeper phenomena in the geometry of Schubert varieties, such as the structure of their resolutions.
• Expansions in the Bruhat lower ideal basis illuminate the underlying order-theoretic features of the Hecke algebra, and the mask description provides a concrete combinatorial realization for basis elements.
• These techniques address questions first raised by Deodhar concerning the existence of explicit positive combinatorial models for the Kazhdan–Lusztig basis, and the resulting methods point toward the potential for new canonical constructions and algorithms.

This comprehensive synthesis exemplifies how combinatorics (heaps, trees, masks), algebra (Hecke algebra, Bruhat order), and geometry (resolutions, cell decompositions) are intertwined in the theory of the Kazhdan–Lusztig basis for cograssmannian permutations (Jones et al., 2010).