Kazhdan-Lusztig Polynomials Research Articles

Kazhdan-Lusztig theory of super type D and quantum symmetric pairs

We reformulate the Kazhdan-Lusztig theory for the BGG category O \mathcal {O} of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category O \mathcal {O} of the ortho-symplectic Lie superalgebra o s p ( 2 m | 2 n ) \mathfrak {osp}(2m|2n) .

Combinatorial invariance of Kazhdan\u2013Lusztig\u2013Vogan polynomials for fixed point free involutions

When mathrm {Sp}(2n,mathbb {C}) acts on the flag variety of mathrm {SL}(2n,mathbb {C}), the orbits are in bijection with fixed point free involutions in the symmetric group S_{2n}. In this case, the associated Kazhdan–Lusztig–Vogan polynomials P_{v,u} can be indexed by pairs of fixed point free involutions vge u, where ge denotes the Bruhat order on S_{2n}. We prove that these polynomials are combinatorial invariants in the sense that if f:[u,w_0]rightarrow [u',w_0] is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then P_{v,u} = P_{f(v),u'} for all vge u.

Algebraic representations and constructible sheaves

I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic p > 0 and Lusztig’s character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg–Mirkovic conjecture) in the spirit of the Kazhdan–Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for GL n ) which should eventually make computations more feasible.

A non-perverse Soergel bimodule in type A

A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan–Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig's conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the existence of a non-perverse parity sheaf in type A.

Kazhdan-Lusztig Polynomials of Thagomizer Matroids

We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.

Kazhdan-Lusztig theory: Origin, development, influence, and some problems

Kazhdan-Lusztig theory is one of the most important progresses in the representation theory of algebraic groups in the past four decades. This theory plays a crucial role in settling up many important problems in Lie theory, for example, the classification of irreducible characters of finite groups of Lie type, and characters of certain irreducible representations in Lie theory. Meanwhile, this theory opens up many dynamic research topics such as Kazhdan-Lusztig polynomials, cells in Coxeter groups, relations among Coxeter groups, intersection cohomology theory and K-theory. In this article we will briefly introduce the origin, development, influence and some unsolved problems of the theory.

The equivariant Kazhdan–Lusztig polynomial of a matroid

We define the equivariant Kazhdan–Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

A combinatorial formula expressing periodic R-polynomials

In 1980, Lusztig introduced the periodic Kazhdan–Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac–Moody algebra at the critical level. The periodic Kazhdan–Lusztig polynomials can be computed by using another family of polynomials, called the periodic R-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic R-polynomials in terms of the “doubled” Bruhat graph associated to a finite Weyl group and a finite root system.

A twisted duality for parabolic Kazhdan–Lusztig R-polynomials

We prove a duality result for the parabolic Kazhdan–Lusztig R-polynomials of a finite Coxeter system. This duality is similar to, but different from, the one obtained in [9]. As a consequence of our duality we obtain an identity between the parabolic Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials of a finite Coxeter system. We also obtain applications to certain modules defined by Deodhar and derive a result that gives evidence in favor of Marietti's combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials.

Character Formulae for Queer Lie Superalgebras and Canonical Bases of Types A/C

For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a Kazhdan-Lusztig conjecture \`a la Brundan is formulated in terms of categorical canonical basis of the $n$th tensor power of the natural representation of the quantum group of type $C$. For the BGG category of $\mathfrak{q}(n)$-modules of congruent non-integral weights, a Kazhdan-Lusztig conjecture is formulated in terms of canonical basis of a mixed tensor of the natural representation and its dual of the quantum group of type $A$. We also establish a character formula for the finite-dimensional irreducible $\mathfrak{q}(n)$-modules of half-integer weights in terms of type $C$ canonical basis of the corresponding $q$-wedge space.

Type A blocks of super category [formula omitted

We show that every block of category O for the general linear Lie superalgebra glm|n(k) is equivalent to some corresponding block of category O for the queer Lie superalgebra qm+n(k). This implies the truth of the Kazhdan–Lusztig conjecture for the so-called type A blocks of category O for the queer Lie superalgebra as formulated by Cheng, Kwon and Wang.

Special matchings in Coxeter groups

Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of any lower Bruhat interval. The results hold in any arbitrary Coxeter group and have also applications in the study of the corresponding parabolic Kazhdan–Lusztig polynomials.

Graded cellularity and the monotonicity conjecture

The graded cellularity of Libedinsky double leaves, which form a basis for the endomorphism ring of the Bott–Samelson–Soergel bimodules, allows us to view the Kazhdan–Lusztig polynomials as graded decomposition numbers. Using this interpretation, we can provide a proof of the monotonicity conjecture for any Coxeter system.

Quantum group of type A and representations of queer Lie superalgebra

We establish a maximal parabolic version of the Kazhdan–Lusztig conjecture [10, Conjecture 5.10] for the BGG category Ok,ζ of q(n)-modules of “±ζ-weights”, where k≤n and ζ∈C∖12Z. As a consequence, the irreducible characters of these q(n)-modules in this maximal parabolic category are given by the Kazhdan–Lusztig polynomials of type A Lie algebras. As an application, closed character formulas for a class of q(n)-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained.

Relative Yangians of Weyl type

Let \(\mathfrak {g}\) be a semisimple Lie algebra. A \(\mathfrak {g}\)-algebra is an algebra R on which \(\mathfrak {g}\) acts by derivations. The diagonal copy of \(\mathfrak {g}\) acting on \(R\otimes U(\mathfrak {g})\) is denoted by \(\mathfrak {k}\). If V is a finite dimensional \(\mathfrak {g}\) module and \(R={\text {End}}V\) (resp. R is the algebra D(V) of differential operators on \(V^*\)) the invariant algebra \((R\otimes U(\mathfrak {g}))^\mathfrak {k}\) is called a relative Yangian (resp. relative Yangian of Weyl type). Extending the essentially complete representation theory of \(({\text {End}}V \otimes U(\mathfrak {g}))^\mathfrak {k}\) inspired by the work of Khoroshkin and Nazarov, a study is made of the representation theory of \(E:=(D(V)\otimes U(\mathfrak {g}))^\mathfrak {k}\). A category \(\mathscr {E}^\star \) of modules for E is introduced and shown to be abelian. Here a number of new techniques have to be introduced because unlike \({\text {End}}V\) the algebra D(V) is infinite dimensional and known to have a rather complicated representation theory. The simples and their projective covers in \(\mathscr {E}^\star \) are described. However an example shows that the simples in \(\mathscr {E}^\star \) do not suffice to exhaust the primitive ideals of E. The description of simple modules for E is reduced to those for D(V). This analysis is valid for certain other \(\mathfrak {g}\)-algebras, notably \(R=U(\mathfrak {g})\). In this it is shown that all the simple modules of \((U\mathfrak {g}) \otimes U(\mathfrak {g}))^\mathfrak {k}\) are finite dimensional and the representation theory of the latter algebra precisely recovers the Kazhdan–Lusztig polynomials. Through Olshanski homomorphisms these results can in principle have applications to the study of Yangians and of twisted Yangians.

Tensor Product Categorifications and the Super Kazhdan–Lusztig Conjecture

We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie super algebra $\mathfrak{gl}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category O for $\mathfrak{gl}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.

Peak algebras, paths in the Bruhat graph and Kazhdan–Lusztig polynomials

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan–Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We point out that, in a certain sense, this formula cannot be simplified.

Special matchings calculate the parabolic Kazhdan–Lusztig polynomials of the universal Coxeter groups

In this paper we prove that the parabolic Kazhdan–Lusztig polynomials and the parabolic R-polynomials of the universal Coxeter group can be computed in a combinatorial way, by using special matchings.

Categorification of a Recursive Formula for Kazhdan–Lusztig Polynomials

We obtain explicit branching rules for graded cell modules and graded simple modules over the endomorphism algebra of a Bott–Samelson bimodule. These rules allow us to categorify a well-known recursive formula for Kazhdan–Lusztig polynomials.

The Primitive Spectrum of a Basic Classical Lie Superalgebra

We prove Conjecture 5.7 in [arXiv:1409.2532], describing all inclusions between primitive ideals for the general linear superalgebra in terms of the Ext1-quiver of simple highest weight modules. For arbitrary basic classical Lie superalgebras, we formulate two types of Kazhdan-Lusztig quasi-orders on the dual of the Cartan subalgebra, where one corresponds to the above conjecture. Both orders can be seen as generalisations of the left Kazhdan-Lusztig order on Hecke algebras and are related to categorical braid group actions. We prove that the primitive spectrum is always described by one of the orders, obtaining for the first time a description of the inclusions. We also prove that the two orders are identical if category O admits `enough' abstract Kazhdan-Lusztig theories. In particular, they are identical for the general linear superalgebra, concluding the proof of the conjecture.