higher order congruence relations on affine moment graphs the subgeneric case abstract we study the structure algebra z of the stable moment graph for the case of the affine root system a 1 the structure algebra z is an algebra over a symmetric algebra and in particular it is a module over a symmetric algebra we study this module structure on z and we construct a basis by setting c equal to zero in z we obtain the module z c 0 this module can be described in terms of the finite root system a 1 and we show that it is determined by a set of certain divisibility relations these relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs and because of this we call them higher order congruence relations

w graph ideals abstract we introduce a concept of a w graph ideal in a coxeter group the main goal of this paper is to describe how to construct a w graph from a given w graph ideal the principal application of this idea is in type a where it provides an algorithm for the construction of w graphs for specht modules

irreducible characters of general linear superalgebra and super duality we develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra including all the finite dimensional modules by directly relating the problem to the classical kazhdan lusztig theory furthermore we prove that certain parabolic bgg categories over the general linear algebra and over the general linear superalgebra are equivalent we also verify a parabolic version of a conjecture of brundan on the irreducible characters in the bgg category of the general linear superalgebra

isomorphisms of hecke modules and parabolic kazhdan lusztig polynomials abstract we define and study some hecke modules which generalize the ones defined by deodhar in 10 and we find isomorphisms between them as consequences we obtain equalities between parabolic kazhdan lusztig polynomials of different coxeter groups in particular we show that any parabolic kazhdan lusztig polynomial equals a parabolic polynomial of a maximal quotient and we find equalities between the parabolic polynomials of quasi minuscule quotients and those of certain maximal quotients of the corresponding affine weyl groups

on the existence of a w graph for an irreducible representation of a coxeter group on montre que toute representation irreductible d un groupe de coxeter fini peut etre fournie par un graphe w

a two sided q analogue of the coxeter complex m cabanes and j rickard showed in 3 that the alvis curtis character duality of a finite group of lie type is induced in non defining characteristic l by a derived equivalence given by tensoring with a bounded complex x and they further conjecture that this derived equivalence should actually be a homotopy equivalence following a suggestion of r kessar we show here for the special case of principal blocks of general linear groups with abelian sylow l subgroups that this is true by an explicit verification relating the complex x to the coxeter complex of the corresponding weyl group

w graph ideals ii abstract in howlett and nguyen 2010 5 the concept of a w graph ideal in a coxeter group was introduced and it was shown how a w graph can be constructed from a given w graph ideal in this paper we describe a class of w graph ideals from which certain kazhdan lusztig left cells arise the result justifies the algorithm as illustrated in howlett and nguyen 2010 5 for the construction of w graphs for specht modules for the hecke algebra of type a

inducing w graphs let be the hecke algebra associated with a coxeter group w many interesting modules can be described using the concept of a w graph as introduced in the influential paper 4 of kazhdan and lusztig in particular kazhdan and lusztig showed that the regular representation of has an associated w graph the purpose of this note is to show that if w j is a parabolic subgroup of w and v is a module for the corresponding hecke algebra j then a w j graph structure for v gives rise to a w graph structure for the induced module j v in the case that w j is the identity subgroup and v has dimension 1 our construction coincides with that given by kazhdan and lusztig for the regular representation for arbitrary j and v of dimension 1 we recover the constructions of couillens 1 and deodhar 3

two sided cells in type b asymptotic case abstract we compute two sided cells of weyl groups of type b for the asymptotic choice of parameters we also obtain some partial results concerning lusztig s conjectures in this particular case

twisted incidence algebras and kazhdan lusztig stanley functions abstract 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

kazhdan lusztig polynomials for hermitian symmetric spaces on presente un schema non recursif pour calculer le polynome de kazhdan lusztig associe a un espace hermitien symetrique classique on donne un tableau des polynomes pour les domaines hermitiens exceptionnels

parabolic kazhdan lusztig polynomials and schubert varieties abstract 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

quasi minimal surfaces of pseudo riemannian space forms with positive relative nullity abstract in this paper we consider surfaces in 4 dimensional pseudo riemannian space forms with index 2 first we obtain some of geometrical properties of such surfaces considering their relative null space then we get classifications of quasi minimal surfaces with positive relative nullity

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

kazhdan lusztig polynomials and character formulae for the lie superalgebra q n abstract 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

classification of spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension spatial surfaces with parallel mean curvature vector play some important roles in general relativity theory of harmonic maps as well as in differential geometry recently spatial surfaces in four dimensional minkowski space time with parallel mean curvature vector were classified by chen and van der veken complete classification of parallel surfaces in 4 dimensional lorentzian space forms tohoku math j 61 1 2009 in this article we completely classify spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension the main result states that there exist 16 families of such surfaces as by product we achieve the complete classification of spatial surfaces with parallel mean curvature vector in minkowski space times of arbitrary dimension

the laurent polynomials my ws in the hecke algebra with unequal parameters abstract let w s be a coxeter system and h the associated hecke algebra with unequal parameters the laurent polynomials m y w s and p y w for y w w and s s play an important role in the representations of h we study the properties of m y w s and p y w the relations among them as well as with the left right and two sided cells of w

some new examples in 1 cohomology this paper gives some new examples in the 1 cohomology theory of finite groups of lie type obtained from both computer calculations and the use of several theoretical results in particular the paper gives the first known examples of 1 cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group the computer calculations were made originally while checking special cases of lusztig s conjecture on characteristic p representations of algebraic groups and we take this opportunity to announce in print some results in that direction they reinforce lusztig s conjecture even in a strong form suggested by kato 2003 published by elsevier science usa

on borwein wiersma decompositions of monotone linear relations monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite not necessarily symmetric matrices in 1970 asplund a monotone convergence theorem for sequences of nonlinear mapping in nonlinear functional analysis american math society providence ri 1970 pp 1 9 studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an irreducible monotone operator in 2007 borwein and wiersma siam j optim 18 2007 pp 946 960 introduced another additive decomposition where the maximal monotone operator is written as the sum of a subdifferential operator and a skew monotone operator both decompositions are variants of the well known additive decomposition of a matrix via its symmetric and skew part this paper presents a detailed study of the borwein wiersma decomposition of a maximal monotone linear relation we give sufficient conditions and characterizations for a maximal monotone linear relation to be borwein wiersma decomposable and show that borwein wiersma decomposability implies asplund decomposability we exhibit irreducible linear maximal monotone operators without full domain thus answering one of the questions raised by borwein and wiersma the borwein wiersma decomposition of any maximal monotone linear relation is made quite explicit in hilbert space

the deodhar decomposition of the grassmannian and the regularity of kp solitons abstract given a point a in the real grassmannian it is well known that one can construct a soliton solution u a x y t to the kp equation the contour plot of such a solution provides a tropical approximation to the solution when the variables x y and t are considered on a large scale and the time t is fixed in this paper we use several decompositions of the grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions first we use the positroid stratification of the real grassmannian in order to characterize the unbounded line solitons in the contour plots at y 0 and y 0 next we use the deodhar decomposition of the grassmannian a refinement of the positroid stratification to study contour plots at t 0 more specifically we index the components of the deodhar decomposition of the grassmannian by certain tableaux which we call go diagrams and then use these go diagrams to characterize the contour plots of solitons solutions when t 0 finally we use these results to show that a soliton solution u a x y t is regular for all times t if and only if a comes from the totally non negative part of the grassmannian

maximally monotone linear subspace extensions of monotone subspaces explicit constructions and characterizations monotone linear relations play important roles in variational inequality problems and quadratic optimizations in this paper we give explicit maximally monotone linear subspace extensions of a monotone linear relation in finite dimensional spaces examples are provided to illustrate our extensions our results generalize a recent result by crouzeix and ocana anaya

screw invariant marginally trapped surfaces in minkowski 4 space a spacelike surface in a lorentzian manifold whose mean curvature vector is lightlike everywhere is called marginally trapped the classification of marginally trapped surfaces in minkowski 4 space which are invariant under a subgroup of the lorentz group that leaves invariant a lightlike direction i e the so called screw invariant surfaces is obtained as corollaries the screw invariant marginally trapped surfaces with harmonic mean curvature vector and with prescribed gaussian curvature are explicitly described

generalisation parabolique des polynomes et des bases de kazhdan lusztig resume on considere un systeme de coxeter un sous systeme parabolique et les deux algebres de hecke correspondantes pour tout caractere de degre un de l algebre de hecke parabolique on considere le module induit de l algebre de hecke parabolique a la grande algebre de hecke chacun de ces modules possede une base standard et deux bases de kazhdan lusztig invariantes par un automorphisme involutif du module considere il en resulte la definition pour chaque caractere lineaire de l algebre de hecke parabolique de deux familles de polynomes de kazhdan lusztig dont on etudie les diverses proprietes existence et unicite symetrie dualite formules de recurrence calcul en utilisant la notion de sous expression distinguee d un element du groupe de coxeter cela generalise une construction faite par deodhar laquelle correspondrait aux deux cas extremes des caracteres indice et signe enfin on etablit des formules donnant l action de la grande algebre de hecke sur les modules induits en termes de bases de kazhdan lusztig copyright 1999 academic press we consider a coxeter system a parabolic subsystem and the two corresponding hecke algebras for each linear character of the parabolic hecke algebra take the induced module from the parabolic hecke algebra to the large hecke algebra for each module there are a standard basis and two kazhdan lusztig bases i e invariant basis by an involution of kazhdan lusztig type so for each linear character of the parabolic algebra we have two families of kazhdan lusztig polynomials for which we compute the classical properties existence and unicity symmetry duality induction formulas and computation of the polynomials using distinguished subexpressions of a coxeter group element this generalizes deodhar s construction which corresponds to the extreme case of sign and index characters finally we give formulas for the action of the hecke algebra on the induced modules in kazhdan lusztig basis terms

parabolic kazhdan lusztig r polynomials for quasi minuscule quotients abstract we give explicit combinatorial formulas for the parabolic kazhdan lusztig r polynomials of the quasi minuscule quotients of the classical weyl groups as an application of our results we obtain explicit combinatorial formulas for certain sums and alternating sums of ordinary kazhdan lusztig r polynomials

standard bases for affine parabolic modules and nonsymmetric macdonald polynomials abstract we establish a connection between degenerate nonsymmetric macdonald polynomials and standard bases and dual standard bases of maximal parabolic modules of affine hecke algebras along the way we prove a weak polynomiality result for coefficients of symmetric and nonsymmetric macdonald polynomials

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

addendum to classification of marginally trapped lorentzian flat surfaces in e24 and its application to biharmonic surfaces abstract due to omission an additional family shall be added to the list of marginally trapped lorentzian flat surfaces in e 2 4 given in theorem 4 1 of b y chen classification of marginally trapped lorentzian flat surfaces in e 2 4 and its application to biharmonic surfaces j math anal appl 340 2008 861 875

translation and shuffling of projectively presentable modules and a categorification of a parabolic hecke module translation and shuffling of projectively presentable modules and a categorification of the parabolic hecke module

duality in parabolic set up for questions in kazhdan lusztig theory in this short article we expand on the ideas developed in de21 viz the so called parabolic analogue of kazhdan lusztig polynomials in that paper we considered two hecke algebra modules corresponding to the two roots 1 and q of the equation u q q 1 which give rise to two sets of polynomials the question of the relationship between those two modules was not considered there since it seemed somewhat of a digression in that context it was noted however that these two sets of polynomials have different recursive relations and so they are different in nature special attention was given to the module corresponding to 1 and it was shown that the coefficients of the polynomials are given by the dimensions of stalks of intersection cohomology sheaves of certain generalized schubert varieties in crystallographic cases recently the author has been working on a combinatorial set up which is used to obtain a closed formula for the kazhdan lusztig polynomials de3 while dealing with the parabolic case one can not use the map j kl sect 11 which gives a dual basis viz ci in case j d see remark 2 5 below at this stage the author came across a preprint by j m douglass do where concrete models for the two hecke algebra modulus are constructed for the parabolic case in a finite coxeter group it is further shown that these two are related to each other via a certain map defined by g lusztig l unfortunately those methods use the finiteness assumption in an essential way the author realized that one may use alternate arguments which do not assume the finiteness as a matter of fact one can now tie up the two hecke algebra modules by a natural duality see theorem 2 1 below which then relates various other

a twisted duality for parabolic kazhdan lusztig r polynomials abstract 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

w graph ideals and duality abstract this paper is the continuation of the work in 14 in that paper we generalized the definition of w graph ideal in the weighted coxeter groups and showed how to construct a w graph from a given w graph ideal in the case of unequal parameters in this paper we study the duality and the full w graph for a given w graph ideal we show that there are two modules associated with a given w graph ideal they are connected by a duality map the full w graph includes all the w graph data determined by the dual and contragredient representations our construction closely parallels that of kazhdan and lusztig 6 10 11 which can be regarded as the special case j it also generalizes the work of couillens 2 deodhar 3 4 and douglass 5 corresponding to the parabolic case

super duality and irreducible characters of ortho symplectic lie superalgebras we formulate and establish a super duality which connects parabolic categories o for the ortho symplectic lie superalgebras and classical lie algebras of bcd types this provides a complete and conceptual solution of the irreducible character problem for the ortho symplectic lie superalgebras in a parabolic category o which includes all finite dimensional irreducible modules in terms of classical kazhdan lusztig polynomials

the decomposition matrices of the brauer algebra over the complex field the brauer algebra was introduced by r brauer in 1937 as a tool in invariant theory the problem of determining the cartan decomposition matrix of the brauer algebra over the complex field has remained open since then here we determine this fundamental invariant

the combinatorial invariance conjecture for parabolic kazhdan lusztig polynomials of lower intervals abstract the aim of this work is to prove a conjecture related to the combinatorial invariance conjecture of kazhdan lusztig polynomials in the parabolic setting for lower intervals in every arbitrary coxeter group this result improves and generalizes among other results the main results of brenti et al 2006 6 marietti 2016 21

bar operators for quasiparabolic conjugacy classes in a coxeter group abstract the action of a coxeter group w on the set of left cosets of a standard parabolic subgroup deforms to define a module m j of the group s iwahori hecke algebra h with a particularly simple form rains and vazirani have introduced the notion of a quasiparabolic set to characterize w sets for which analogous deformations exist a motivating example is the conjugacy class of fixed point free involutions in the symmetric group deodhar has shown that the module m j possesses a certain antilinear involution called the bar operator and a certain basis invariant under this involution which generalizes the kazhdan lusztig basis of h the well known significance of this basis in representation theory makes it natural to seek to extend deodhar s results to the quasiparabolic setting in general the obstruction to finding such an extension is the existence of an appropriate quasiparabolic analogue of the bar operator in this paper we consider the most natural definition of a quasiparabolic bar operator and develop a theory of quasiparabolic kazhdan lusztig bases under the hypothesis that such a bar operator exists giving content to this theory we prove that a bar operator in the desired sense does exist for quasiparabolic w sets given by twisted conjugacy classes of twisted involutions finally we prove several results classifying the quasiparabolic conjugacy classes in a coxeter group

parabolic kazhdan lusztig polynomials for hermitian symmetric pairs we study the parabolic kazhdan lusztig polynomials for hermitian symmetric pairs in particular we show that these polynomials are always either zero or a monic power of q and that they are combinatorial invariants

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and is a primitive l th root of unity with l larger than the coxeter number of g the calculation employs the prior calculations and techniques of ostrik and of nakano parshall and vella as well as deep results involving the validity of the lusztig character formula for quantum groups and the positivity of parabolic kazhdan lusztig polynomials for the affine weyl group analogous support variety calculations are provided for the first frobenius kernel g 1 of a reductive algebraic group scheme g defined over the prime field f p

parallel surfaces in lorentzian three manifolds admitting a parallel null vector field we classify parallel surfaces in lorentzian three manifolds admitting a parallel null vector field some families of semi parallel surfaces are also described

quantum weyl reciprocity for cohomology we study the relationship between the cohomology of q schur algebras and hecke algebras in type a a key tool which has independent interest involves a specific hecke algebra resolution introduced in a special case by deodhar this resolution is somewhat akin to the koszul resolution for lie algebra cohomology in that it has an explicit description yet applies for all modules an application of the properties of this resolution establishes a quantum weyl reciprocity connecting in a range of degrees the cohomology of the q schur algebra with that of the hecke algebra this work also provides independently of the resolution cohomological inequalities which apply in all degrees and which sometimes can be related to kazhdan lusztig polynomials finally we extend to cohomology a decomposition number method of erdmann and illustrate its usefulness in computations this method enables us to exhibit some properties of symmetric group cohomology which if true would imply the sl n version of lusztig s characteristic p character conjecture

the jones wenzl idempotent of a generalized temperley lieb algebra abstract we define an idempotent of the temperley lieb algebra of any finite coxeter system which reduces in type a to the known jones wenzl idempotent we give for any pair of parabolic subgroups a recursive formula generalizing the well known one this approach gives a wide class of recursive formulas for the classical jones wenzl idempotent we also compute explicitly the coefficient corresponding to the maximal element of any minuscule quotient when the idempotent is expressed in the basis of fully commutative elements

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

kazhdan lusztig r polynomials for pircons abstract the purpose of this work is to provide a common combinatorial framework for some of the analogues and generalizations of kazhdan lusztig r polynomials that have appeared since the introduction of these remarkable polynomials e g parabolic kazhdan lusztig r polynomials kazhdan lusztig r polynomials of zircons and kazhdan lusztig vogan polynomials for fixed point free involutions

closed product formulas for extensions of generalized verma modules we give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of g p generalized verma modules in the cases when g p corresponds to an indecomposable classic hermitian symmetric pair the formulas imply that these dimensions are combinatorial invariants we also discuss how these polynomials defined by shelton are related to the parabolic r polynomials introduced by deodhar

the kernel average for two convex functions and its application to the extension and representation of monotone operators we provide and analyze an average for two convex functions based on a kernel function it covers several known averages such as the arithmetic average epigraphical average and the proximal average when applied to the fitzpatrick function and the conjugate of fitzpatrick function associated with a monotone operator our average produces an autoconjugate also known as selfdual lagrangian which can be used for finding an explicit maximal monotone extension of the given monotone operator this completely settles one of the open problems posed by fitzpatrick in the setting of reflexive banach spaces

fully commutative elements in the weyl and affine weyl groups abstract let w be a weyl or an affine weyl group and let w c be the set of fully commutative elements in w we associate each w w c to a digraph g w by using g w we give a graph theoretic description for lusztig s a function on w c and describe explicitly all the distinguished involutions of w c the results verify two conjectures in our case one was proposed by myself in adv sci china math 3 1990 79 98 conjecture 8 10 and the other by lusztig in t asai et al open problems in algebraic groups in r hotta ed problems from the conference on algebraic groups and representations katata august 29 september 3 1983

the weighted universal coxeter group and some related conjectures of lusztig abstract lusztig proposed 15 conjectures for any weighted coxeter groups in this study we explicitly describe kazhdan lusztig cells distinguished involutions and the function a for any weighted universal coxeter group we also verify these conjectures for this group

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

special matchings calculate the parabolic kazhdan lusztig polynomials of the universal coxeter groups abstract 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

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

categorification of the temperley lieb category tangles and cobordisms via projective functors to each generic tangle projection from the three dimensional real vector space onto the plane we associate a derived endofunctor on a graded parabolic version of the bernstein gel fand category mathcal o we show that this assignment is up to shifts invariant under tangle isotopies and reidemeister moves and defines therefore invariants of tangles the occurring functors are defined via so called projective functors the first part of the paper deals with the indecomposability of such functors and their connection with generalised temperley lieb algebras which are known to have a realisation via decorated tangles the second part of the paper describes a categorification of the temperley lieb category and proves the main conjectures of bfk moreover we describe a functor from the category of 2 cobordisms into a category of projective functors

parabolic kazhdan lusztig r polynomials for tight quotients of the symmetric groups abstract we give explicit closed combinatorial formulas for the parabolic kazhdan lusztig r polynomials of the tight quotients of the symmetric groups we give two formulations of our result one in terms of permutations and one in terms of motzkin paths as an application of our results we obtain explicit closed combinatorial formulas for certain sums and alternating sums of ordinary kazhdan lusztig r polynomials

spatial and lorentzian surfaces in robertson walker space times let l14 f c i fs gfc be a robertson walker space time which does not contain any open subset of constant curvature in this paper we provide a general study of nondegenerate surfaces in l14 f c first we prove the nonexistence of marginally trapped surfaces with positive relative nullity then we classify totally geodesic submanifolds finally we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector

kazhdan lusztig combinatorics in the moment graph setting abstract motivated by a question on the graded rank of the stalks of the canonical sheaf on a bruhat graph we lift some equalities concerning parabolic kazhdan lusztig polynomials to this moment graph setting our proofs hold also in positive characteristic under some technical assumptions

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry