Kazhdan-Lusztig Polynomials Research Articles

Schubert polynomials, Kazhdan–Lusztig basis and characters

A combinatorial formula for the characters of the homogeneous components of the coinvariant algebra is given. The formula is proved by considering the action of the simple reflections on the Schubert polynomials basis of this algebra. In the symmetric group case, the formula is equivalent to a combinatorial rule for decomposing the homogeneous components into irreducible representations. The proof of the equivalence involves permutation statistics and Kazhdan–Lusztig theory. The formula is very similar to an analogous one for Kazhdan–Lusztig representations of these groups.

On matrix coefficients of the Satake isomorphism: complements to the paper of M. Rapoport

We consider the matrix for the Satake isomorphism with respect to natural bases. We give a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers. We also relate the matrix coefficients to Kazhdan–Lusztig polynomials and to Bernstein functions.

An Abstract Model for Bruhat Intervals

We introduce two constructions, which we call extension and shift, that will construct new posets from old. These constructions are defined in terms of labellings of the elements of the poset by signs $+$, $-$, that axiomatize the ones induced by the left multiplication by a generator $s$ on an interval in the Bruhat ordering of a Coxeter group; in that case the label is $+$ when the length goes up, $-$ when it goes down. Starting from the one-element poset, these constructions yield a class of Eulerian posets containing, but not limited to, all intervals in (finitely generated) Coxeter groups. We explain how analogues of Kazhdan-Lusztig polynomials may be defined for such a poset in terms of a given construction of the poset. It is hoped that for a restricted set of constructions, the result will be independent of the choice of construction. We prove this for posets that do not contain dihedral subintervals of length more than two.

Explicit Formulae for Some Kazhdan-Lusztig Polynomials

We consider the Kazhdan-Lusztig polynomials P u,v (q) indexed by permutations u, v having particular forms with regard to their monotonicity patterns. The main results are the following. First we obtain a simplified recurrence relation satisfied by P u,v (q) when the maximum value of v ∈ Sn occurs in position n − 2 or n − 1. As a corollary we obtain the explicit expression for Pe,3 4 ... n 1 2(q) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for Pe, 3 4 ... (n − 2) n (n − 1) 1 2(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for P u,v (q) under hypotheses similar to those of the main results.

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tohoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.

Twisted Incidence Algebras and Kazhdan–Lusztig–Stanley Functions

We introduce a new multiplication in the incidence algebra of a partially ordered set and study the resulting algebra. As an application of the properties of this algebra we obtain a combinatorial formula for the Kazhdan–Lusztig–Stanley functions of a poset. As special cases this yields new combinatorial formulas for the parabolic and inverse parabolic Kazhdan–Lusztig polynomials, for the generalized (toric) h-vector of an Eulerian poset and for the Lusztig–Vogan polynomials.

Formula omitted] Algebras with three simple modules

We classify O algebras with three simple modules by quivers and relations, and then determine their representation types. The exact Borel subalgebras, the Kazhdan–Lusztig–Vogan polynomials, and the Ringel duals of these algebras are also studied.

Strong Exact Borel Subalgebras of Quasi-hereditary Algebras and Abstract Kazhdan–Lusztig Theory

Strong exact Borel subalgebras and strong Δ-subalgebras are shown to exist for quasi-hereditary algebras which possess exact Borel subalgebras and Δ-subalgebras. This implies that the algebras associated with blocks of category O have strong exact Borel subalgebras and strong Δ-subalgebras. The structure of these subalgebras is shown to be closely related to abstract Kazhdan–Lusztig theory. The main technical tool in this paper is a construction which has an exact Borel subalgebras (of a given quasi-hereditary algebra) as input and a strong exact Borel subalgebra as output. From this result, Morita invariance of the existence of exact Borel subalgebras is derived.

Drinfeld Functor and Finite-Dimensional Representations of Yangian

We extend the results of Drinfeld on Drinfeld functor to the case l>n. We present the character of finite-dimensional representations of the Yangian Y(sl_n) in terms of the Kazhdan-Lusztig polynomials as a consequence.

Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups

To each polynomial P P with integral nonnegative coefficients and constant term equal to 1 1 , of degree d d , we associate a certain pair of elements ( y , w ) (y,w) in the symmetric group S n S_n , where n = 1 + d + P ( 1 ) n = 1 + d + P(1) , such that the Kazhdan-Lusztig polynomial P y , w P_{y,w} equals P P . This pair satisfies ℓ ( w ) − ℓ ( y ) = 2 d + P ( 1 ) − 1 \ell (w) - \ell (y) = 2d + P(1) - 1 , where ℓ ( w ) \ell (w) denotes the number of inversions of w w .

A Construction of Weighted Parabolic Kazhdan–Lusztig Polynomials

A Construction of Weighted Parabolic Kazhdan–Lusztig Polynomials

Construction de polynômes de Kazhdan-Lusztig arbitraires

To each polynomial P with integral non-negative coefficients and constant term equal to 1, of degree d, we associate a pair of elements (y,w) in the symmetric group S n, where n = 1 + d + P(l), for which we prove that the Kazhdan-Lusztig polynomial P y,w equals P. This pair satisfies ℓ(w) - ℓ(y) = 2d + P(1) - 1, where ℓ(w) denotes the number of inversions of w.

Characters and composition factor multiplicities for the Lie superalgebra gl(m/n).

The multiplicities aλ μ of simple modules Lμ in the composition series of Kac modules V lambda for the Lie superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ) were described by Serganova, leading to her solution of the character problem for \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ). In Serganova's algorithm all μ with nonzero aλ μ are determined for a given λ this algorithm, turns out to be rather complicated. In this Letter, a simple rule is conjectured to find all nonzero aλ μ for any given weight μ. In particular, we claim that for an r-fold atypical weight μ there are 2r distinct weights λ such that aλ μ = 1, and aλ μ = 0 for all other weights λ. Some related properties on the multiplicities aλ μ are proved, and arguments in favour of our main conjecture are given. Finally, an extension of the conjecture describing the inverse of the matrix of Kazhdan–Lusztig polynomials is discussed.

Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials

We present a fast version of the algorithm of Lascoux, Leclerc, and Thibon for the lower global crystal base for the Fock representation of quantum affine sl_n. We also show that the coefficients of the lower global crystal base coincide with certain affine Kazhdan-Lusztig polynomials. It is known that the coefficients of the global crystal base are q-analogues of decomposition numbers for Specht modules of the Hecke algebra of type A_n, and that the coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of decomposition numbers for tilting modules for quantum sl_k. Thus our algorithm allows fast computation of these decomposition numbers.

Kazhdan-Lusztig polynomials and canonical basis

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the groupSn coincide with the coefficients of the canonical basis innth tensor power of the fundamental representation of the quantum groupUq\(\mathfrak{s}\mathfrak{l}\)k. We also use known results about canonical bases forUq\(\mathfrak{s}\mathfrak{l}\)2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schutzenberger and Zelevinsky.

Upper and Lower Bounds for Kazhdan–Lusztig Polynomials

We give upper and lower bounds for the Kazhdan–Lusztig polynomials of any Coxeter groupW. IfWis finite we prove that, for any k≥0, thekth coefficient of the Kazhdan–Lusztig polynomial of two elementsu,vofWis bounded from above by a polynomial (which depends only onk) inl(v) −l(u). In particular, this implies the validity of Lascoux–Schutzenberger's conjecture for all sufficiently long intervals, and gives supporting evidence in favour of the Dyer–Lusztig conjecture.

Cells andq-Schur algebras

This paper shows how the Kazhdan-Lusztig theory of cells can be directly applied to establish the quasi-heredity ofq-Schur algebras. The application arises because of a very strong homological property enjoyed by certain cell filtrations forq-permutation modules.

Kazhdan–Lusztig Polynomials of Parabolic Type

Kazhdan–Lusztig Polynomials of Parabolic Type

Lattice paths and Kazhdan-Lusztig polynomials

The purpose of this paper is to present a new non-recursive combinatorial formula for the Kazhdan-Lusztig polynomials of a Coxeter group W W . More precisely, we show that each directed path in the Bruhat graph of W W has a naturally associated set of lattice paths with the property that the Kazhdan-Lusztig polynomial of u , v u,v is the sum, over all the lattice paths associated to all the paths going from u u to v v , of ( − 1 ) Γ ≥ 0 + d + ( Γ ) q ( l ( v ) − l ( u ) + Γ ( l ( Γ ) ) ) / 2 (-1)^{\Gamma _{\ge 0}+d_+(\Gamma )}q^{(l(v)-l(u)+\Gamma (l(\Gamma )))/2} where Γ ≥ 0 , d + ( Γ ) \Gamma _{\ge 0}, d_+(\Gamma ) , and Γ ( l ( Γ ) ) \Gamma (l(\Gamma )) are three natural statistics on the lattice path.

The composition series of modules induced from Whittaker modules

We study a category of representations over a semisimple Lie algebra, which contains category $ \cal O $ as well as the so-called Whittaker modules, and prove a generalization of the Kazhdan-Lusztig conjectures in this context.