Kazhdan-Lusztig Polynomials Research Articles

Maximal singular loci of Schubert varieties in $SL(n)/B$

Schubert varieties in the flag manifold SL(n)/B play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety X w is nonsingular if and only if w avoids the patterns 4231 and 3412. They also gave a conjectural description of the singular locus of X w . In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety X w for any element w ∈ G n . In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from w by a cycle depending naturally on a 4231 or 3412 pattern in w. Our description of the irreducible components is computationally more efficient (O(n 6 )) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.

The distinguished involutions with a-value n2−3 n+3 in the Weyl group of type Dn

Let ( W, S) be a Weyl group and H its associated Hecke algebra. Let A= Z[u,u −1] be the Laurent polynomial ring. Kazhdan and Lusztig [Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] introduced two A -bases { T w } w∈ W and { C w } w∈ W for the Hecke algebra H associated to W. Let Y w =∑ y⩽ w u l( w)− l( y) T y . Then { Y w } w∈ W is also an A -base for the Hecke algebra. In this paper we give an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of Y x 's in the Hecke algebra of type D n . In fact, this gives also an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of T x 's in the Hecke algebra of type D n . Thus we describe also explicitly the Kazhdan–Lusztig polynomials for certain elements of the Weyl group. We study also the joint relation among some elements in W and some distinguished involutions with a-value n 2−3 n+3 in the Weyl group of type D n .

Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra [formula omitted

We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra q(n) and formulate a precise conjecture for the other irreducible characters in category O . The guiding principle is that the representation theory of q(n) should be described by Lusztig's canonical basis for “tensor space” over the quantum group of type b ∞ .

On the combinatorics of B× B-orbits on group compactifications

It is shown that there is an order isomorphism ϕ′ from the poset V of B× B-orbits on the wonderful compactification of a semi-simple adjoint group G with Weyl group W to an interval in reverse Chevalley–Bruhat order on a non-canonically associated Coxeter group W (in general neither finite nor affine). Moreover, ϕ′ preserves the corresponding Kazhdan–Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan–Lusztig polynomials for the analogues of V for general Coxeter groups W is completed by reducing it by a similar order isomorphism to known results involving a “twisted” Chevalley–Bruhat order on W .

Rigidity of Schubert closures and invariance of Kazhdan–Lusztig polynomials

Let [ e, w], [ e, w′] be intervals from the origin in two Coxeter groups W, W′. We prove that any poset isomorphism ϕ : [e,w]→[e,w′] preserves Kazhdan–Lusztig polynomials, in the sense that P ϕ( x), ϕ( y) = P x, y for any x⩽ y in [ e, w], in the case where one of the two groups W, W′ has the property that the Coxeter graph of each of its irreducible constituents is either a tree, or affine of type A ̃ n ; in particular, the result holds for all finite or affine Coxeter groups. We obtain this result as a consequence of an explicit description of all the poset isomorphisms from [ e, w] to [ e, w′].

On the topology of components of some Springer fibers and their relation to Kazhdan–Lusztig theory

We describe the irreducible components of Springer fibers for hook and two-row nilpotent elements of gl n( C) as iterated bundles of flag manifolds and Grassmannians. We then relate the topology (in particular, the intersection homology Poincaré polynomials) of the pairwise intersections of these components with the inner products of the Kazhdan–Lusztig basis elements of irreducible representations of the rational Iwahori–Hecke algebra of type A corresponding to the hook and two-row Young shapes.

P-kernels, IC bases and Kazhdan–Lusztig polynomials

In [J. Amer. Math. Soc. 5 (1992) 805–851] Stanley introduced the concept of a P-kernel for any locally finite partially ordered set P. In [Proc. Sympos. Pure Math., Vol. 56, AMS, 1994, pp. 135–148] Du introduced, for any set P, the concept of an IC basis. The purpose of this article is to show that, under some mild hypotheses, these two concepts are equivalent, and to characterize, for a given Coxeter group W, partially ordered by Bruhat order, the W-kernel corresponding to the Kazhdan–Lusztig basis of the Hecke algebra of W. Finally, we show that this W-kernel factorizes as a product of other W-kernels, and that these provide a solution to the Yang–Baxter equations for W.

Kazhdan–Lusztig and R-polynomials, Young’s lattice, and Dyck partitions

We give explicit combinatorial product formulas for the maximal parabolic Kazhdan-Lusztig and R-polynomials of the symmetric group. These formulas imply that these polynomials are combinatorial invariants, and that the Kazhdan-Lusztig ones are nonnegative. The combinatorial formulas are most naturally stated in terms of Young's lattice, and the one for the Kazhdan-Lusztig polynomials depends on a new class of skew partitions which are closely related to Dyck paths.

Some closed formulas for canonical bases of Fock spaces

We give some closed formulas for certain vectors of the canonical bases of the Fock space representation ofUv(sln)U_v(\mathfrak {sl}_n). As a result, a combinatorial description of certain parabolic Kazhdan-Lusztig polynomials for affine typeAAis obtained.

On the center of affine Hecke algebras of type A

We give an explicit expression for the central elements of affine Hecke algebras of type A in the Coxeter presentation, in terms of (parabolic) affine Kazhdan-Lusztig polynomials. Our approach is based on a version of quantum affine Schur-Weyl duality involving the Hall algebra of a cyclic quiver, on the theory of canonical bases and on the higher-level q-deformed Fock spaces recently introduced by Uglov.

Enumerative and Combinatorial Properties of Dyck Partitions

The purpose of this paper is to study the combinatorial and enumerative properties of a new class of (skew) integer partitions. This class is closely related to Dyck paths and plays a fundamental role in the computation of certain Kazhdan–Lusztig polynomials of the symmetric group related to Young's lattice. As a consequence of our results, we obtain some new identities for these polynomials.

Ringel duality and Kazhdan–Lusztig theory

Ringel duality exhibits a symmetry for quasi-hereditary algebras, which, in particular, is of interest for blocks of the BGG-category O and for Schur algebras of classical groups. This symmetry is used to phrase (Kazhdan-)Lusztig conjecture in terms of maps between tilting modules and also in terms of composition factors occuring in certain layers of good or cogood filtrations of tilting modules. The conditions make sense for centralizer subalgebras as well where they can be formulated in terms of non-existence of certain uniserial submodules. Hence, the validity of (Kazhdan-)Lusztig conjecture is equivalent to a 'regularity' condition on the structure of tilting modules.

Parabolic Kazhdan–Lusztig Polynomials and Schubert Varieties

We shall give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac–Moody Lie algebras in terms of parabolic Kazhdan–Lusztig polynomials introduced by Deodhar.

The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of ‘cell modules’ for the extended affine Hecke algebra of type A which are parametrised by SLn(ℂ)-conjugacy classes of pairs (s, N), where s ∈ SLn(ℂ) is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q2N. When q2≠−1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q2=−1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the ‘standard modules’ earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]

Closed Product Formulas for CertainR-polynomials

R -polynomials get their importance from the fact that they are used to define and compute the Kazhdan–Lusztig polynomials, which have applications in several fields. Here we give a closed product formula for certainR -polynomials valid for every Coxeter group. This result implies a conjecture due to F. Brenti about the symmetric groups.

Computing Kazhdan-Lusztig Polynomials for Arbitrary Coxeter Groups

Let (W, S) be an arbitrary Coxeter system, y ∊ S*. We describe an algorithm which will compute, directly from y and the Coxeter matrix of W, the interval from the identity to y in the Bruhat ordering, together with the (partially defined) left and right actions of the generators. This provides us with exactly the data that are needed to compute the Kazhdan-Lusztig polynomials P x, z , x ≤ z ≤ y. The correctness proof of the algorithm is based on a remarkable theorem due to Matthew Dyer

A path algorithm for affine Kazhdan-Lusztig polynomials

We develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A, which in turn is a faster version of an algorithm due to Lascouz, Leclerc and Thibon (proposed in the setting of Hecke algebras of type A, at roots of unity.)

Filtrations on G 1 T -Modules

Let G be an almost simple algebraic group defined over Fp for some prime p. Denote by G1 the first Frobenius kernel in G and let T be a maximal torus. In this paper we study certain Jantzen type filtrations on various modules in the representation theory of G1T. We have such filtrations on the baby Verma modules Zλ, where λ is a character of T. They are obtained via a certain deformation of the natural homomorphism from Zλ into its contravariant dual Zλτ. Using the same deformation we construct for each projective G1T-module Q a filtration of the vector space . We then prove that this filtration may also be described in terms of the above-mentioned homomorphism Z(λ) → Z(λ)τ and this leads us to a sum formula for our filtrations. When Q is indecomposable with highest weight in the bottom alcove (with respect to some special point) we are able to compute the filtrations on Fλ(Q) explicitly for all λ. This is then the starting point of an induction which proceeds via wall crossings to higher alcoves. If our filtrations behave as expected under such wall crossings then we obtain a precise relation betweenthe dimensions of the layers in the filtrations of Fλ(Q) for an arbitrary indecomposable projective Q and the coefficients in the corresponding Kazhdan–Lusztig polynomials. We conclude the paper by proving that the above results in the G1T theory have some analogues in the representation theory of G (where, however, we have to work with representations of bounded highest weights) and the corresponding theory for quantum groups at roots of unity. These results extend previous work by the first author. 2000 Mathematics Subject Classification: 20G05, 20G10, 17B37.

Some Properties of Inverse Weighted Parabolic Kazhdan–Lusztig Polynomials

Some Properties of Inverse Weighted Parabolic Kazhdan–Lusztig Polynomials

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

In (Deodhar, i>Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials i>Px,w in the case where i>W is any Coxeter group. We explicitly describe the combinatorics in the case where W=\hbox{\ca}_n (the symmetric group on i>n letters) and the permutation i>w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for i>w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincare polynomial of the intersection cohomology of the Schubert variety corresponding to i>w is (1+q)^{l(w)} if and only if i>w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety i>Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of i>Xw when i>w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (i>Bn, i>F4, i>G2).