isoperimetric control of the steklov spectrum abstract we prove that the normalized steklov eigenvalues of a bounded domain in a complete riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain consequently the normalized steklov eigenvalues of a bounded domain in euclidean space hyperbolic space or a standard hemisphere are uniformly bounded above on a compact surface with boundary we obtain uniform bounds for the normalized steklov eigenvalues in terms of the genus we also establish a relationship between the steklov eigenvalues of a domain and the eigenvalues of the laplace beltrami operator on its boundary hypersurface

regular cell complexes in total positivity fomin and shapiro conjectured that the link of the identity in the bruhat stratification of the totally nonnegative real part of the unipotent radical of a borel subgroup in a semisimple simply connected algebraic group defined and split over mathbb r is a regular cw complex homeomorphic to a ball the main result of this paper is a proof of this conjecture this completes the solution of the question of bernstein of identifying regular cw complexes arising naturally from representation theory having the lower intervals of bruhat order as their closure posets a key ingredient is a new criterion for determining whether a finite cw complex is regular with respect to a choice of characteristic maps it most naturally applies to images of maps from regular cw complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology

embeddings of submanifolds and normal bundles this paper studies the embeddings of a complex submanifold s inside a complex manifold m in particular we are interested in comparing the embedding of s in m with the embedding of s as the zero section in the total space of the normal bundle ns of s in m we explicitly describe some cohomological classes allowing to measure the difference between the two embeddings in the spirit of the work by grauert griffiths and camacho movasati and sad we are also able to explain the geometrical meaning of the separate vanishing of these classes our results hold for any codimension but even for curves in a surface we generalize previous results due to laufert and camacho movasati and sad

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

the combinatorics of twisted involutions in coxeter groups the open intervals in the bruhat order on twisted involutions in a coxeter group are shown to be pl spheres this implies results conjectured by f incitti and sharpens the known fact that these posets are gorenstein over z 2 we also introduce a boolean cell complex which is an analogue for twisted involutions of the coxeter complex several classical coxeter complex properties are shared by our complex when the group is finite it is a shellable sphere shelling orders being given by the linear extensions of the weak order on twisted involutions furthermore the h polynomial of the complex coincides with the polynomial counting twisted involutions by descents in particular this gives a type independent proof that the latter is symmetric

group actions on stanley reisner rings and invariants of permutation groups abstract let h be a group of permutations of x 1 x n and let q h x 1 x 2 x n denote the ring of h invariant polynomials in x 1 x 2 x n with rational coefficients combinatorial methods for the explicit construction of free bases for q h x 1 x 2 x n as a module over the symmetric polynomials are developed the methods are developed by studying the action of the symmetric group on the stanley reisner ring of the subset lattice some general results are also obtained by studying the action of a coxeter group on the stanley reisner ring of the corresponding coxeter complex in the case of a weyl group a purely combinatorial construction of certain invariants first considered by r steinberg topology 14 1975 173 177 is obtained some applications to representation theory are also included

pattern avoidance and rational smoothness of schubert varieties abstract letwbe an element of the weyl groupsn and letxwbe the schubert variety associated towin the flag manifoldsln c b lakshmibai and sandhya showed thatxwis smooth if and only ifwavoids the patterns 4231 and 3412 using two tests for rational smoothness from carrell and peterson we show that rational smoothness ofxwis characterized by pattern avoidance for typesbandcas well a key step in the proof of this result is a sequence of rules for factoring the poincare polynomials for the cohomology ring ofxw generalizing the work of gasharov patterns can also be used to characterize actual smoothness of schubert varieties

discrete morse theory for totally non negative flag varieties abstract in a seminal 1994 paper lusztig 1994 26 lusztig extended the theory of total positivity by introducing the totally non negative part g p 0 of an arbitrary generalized partial flag variety g p he referred to this space as a remarkable polyhedral subspace and conjectured a decomposition into cells which was subsequently proven by the first author rietsch 1998 33 in williams 2007 40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron by conjecturing it to be a regular cw complex that is homeomorphic to a closed ball in this article we use discrete morse theory to prove this conjecture up to homotopy equivalence explicitly we prove that the boundaries of the cells are homotopic to spheres and the closures of cells are contractible the latter part generalizes a result of lusztig s 1998 28 that g p 0 the closure of the top dimensional cell is contractible concerning our result on the boundaries of cells even the special case that the boundary of the top dimensional cell g p 0 is homotopic to a sphere is new for all g p other than projective space

determining anisotropic real analytic conductivities by boundary measurements if an electrical potential is applied to the surface of a solid body the current flux across the surface depends on the conductivity in the interior of the body we want to consider the inverse problem of determining the conductivity by these boundary measurements

subword complexes in coxeter groups abstract let be a coxeter system an ordered list of elements in and an element in determine a subword complex as introduced in knutson and miller ann of math 2 2003 to appear subword complexes are demonstrated here to be homeomorphic to balls or spheres and their hilbert series are shown to reflect combinatorial properties of reduced expressions in coxeter groups two formulae for double grothendieck polynomials one of which appeared in fomin and kirillov proceedings of the sixth conference in formal power series and algebraic combinatorics dimacs 1994 pp 183 190 are recovered in the context of simplicial topology for subword complexes some open questions related to subword complexes are presented

generalized quotients in coxeter groups for w s a coxeter group we study sets of the form w v w e w i l wv 1 w i v for all v e v where v c w such sets w v here called generalized quotients are shown to have much of the rich combinatorial structure under bruhat order that has previously been known only for the case when v c s i e for minimal coset representatives modulo a parabolic subgroup we show that bruhat intervals in w v for general v c w are lexicographically shellable the mobius function on w v under bruhat order takes values in 1 0 1 for finite groups w generalized quotients are the same thing as lower intervals in the weak order this is however in general not true connections with the weak order are explored and it is shown that w v is always a complete meet semilattice and a convex order ideal as a subset of w under weak order descent classes di w e w il1 ws 1 w xt s e i for all s e s i c s are also analyzed using generalized quotients it is shown that each descent class as a poset under bruhat order or weak order is isomorphic to a generalized quotient under the corresponding ordering the latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics for instance the set of standard young tableaux of a fixed shape or the set of linear extensions of a rooted forest suitably interpreted form generalized quotients we prove a factorization result for the quotients that come from rooted forests which shows that algebraically these quotients behave as a system of minimal coset representatives of a subset which is in general not a subgroup we also study the rank generating function for certain quotients in the symmetric group

combinatorial properties of the kazhdan lusztigr polynomials forsn abstract we point out a deep and surprising connection between the kazhdan lusztigr polynomials forsnand the enumeration and combinatorics of increasing subsequences in permutations this leads to a simple combinatorial recurrence and to several new closed formulas for these polynomials

index theorems for holomorphic self maps let m be a complex manifold and s subset m a possibly singular subvariety of m let f colon m to m be a holomorphic map such that f restricted to s is the identity we show that one can associate to f a holomorphic section x f of a sheaf related to the embedding of s in m and that such a section reads the dynamical behavior of f along s in particular we prove that under generic hypotheses the canonical section x f induces a holomorphic action in the sense of bott on the normal bundle of the regular part of s in m and this allows to obtain for holomorphic self maps with non isolated fixed points index theorems similar to camacho sad baum bott and variation index theorems for holomorphic foliations finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self maps of surfaces with a compact curve of fixed points

a lookup conjecture for rational smoothness the purpose of this paper is to present a new combinatorial criterion for rational smooth ness of a point in a schubert variety the main result proves the criterion for schubert varieties associated to partial flag varieties of the form g p where g is semisimple of classical type and p is a maximal parabolic subgroup in the case where g p is of cominuscule type not necessarily clas sical we prove a stronger result we conjecture that the criterion is valid for all schubert varieties the paper also contains some general results related to rational smoothness in schubert varieties 1 introduction the purpose of this paper is to present a new combinato rial criterion for rational smoothness of a point in a schubert variety the main result proves the criterion for schubert varieties associated to certain partial flag varieties we conjecture that it is true in general the paper also contains some general results related to rational smoothness in schubert varieties we now explain our criterion in more detail let g be a complex semisimple

discrete morse functions from lexicographic orders this paper shows how to construct a discrete morse function with a relatively small number of critical cells for the order complex of any finite poset with o and i from any lexicographic order on its maximal chains specifically if we attach facets according to the lexicographic order on maximal chains then each facet contributes at most one new face which is critical and at most one betti number changes facets which do not change the homotopy type also do not contribute any critical faces dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets as one application the mobius function may be computed as the alternating sum of morse numbers the above construction enables us to prove that the poset n s of partitions of a set 1 1 k k with repetition is homotopy equivalent to a wedge of spheres of top dimension when a is a hook shaped partition it is likely that the proof may be extended to a larger class of a and perhaps to all a despite a result of ziegler 1986 which shows that n s is not always cohen macaulay

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

submanifolds with splitting tangent sequence a theorem of van de ven states that a projective submanifold of complex projective space whose holomorphic tangent bundle sequence splits holomorphically is necessarily a linear subspace note that the sequence always splits differentiably but in general not holomorphically we are interested in generalizations to the case when the ambient space is a homogeneous manifold different from projective space quadrics grassmannians or abelian manifolds for example split submanifolds are closely related to totally geodesic submanifolds

a criterion for smoothness of schubert varieties insp 2n b abstract in this paper we give a criterion for a schubert variety x w in sp 2 n b to be smooth the criterion is given in terms of the permutation giving w

irreducible representations of semisimple algebraic groups and supersolvable lattices abstract let be the cross section lattice of an irreducible representation of a semisimple algebraic group certain combinatorial properties of are studied supersolvable s are determined in terms of dynkin diagrams the characteristic polynomial of a supersolvable is computed

the steklov and laplacian spectra of riemannian manifolds with boundary abstract given two compact riemannian manifolds m 1 and m 2 such that their respective boundaries 1 and 2 admit neighbourhoods 1 and 2 which are isometric we prove the existence of a constant c such that k m 1 k m 2 c for each k n the constant c depends only on the geometry of 1 2 this follows from a quantitative relationship between the steklov eigenvalues k of a compact riemannian manifold m and the eigenvalues k of the laplacian on its boundary our main result states that the difference k k is bounded above by a constant which depends on the geometry of m only in a neighbourhood of its boundary the proofs are based on a pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces in several situations the constant c is given explicitly in terms of bounds on the geometry of 1 2

a combinatorial formula for kazhdan lusztig polynomials in their fundamental paper 14 kazhdan and lusztig defined for every coxeter group w a family of polynomials indexed by pairs of elements of w which have become known as the kazhdan lusztig polynomials of w see e g 12 chap 7 these polynomials are intimately related to the bruhat order of w and to the algebraic geometry of schubert varieties and have proven to be of fundamental importance in representation theory our aim in this work is to give a simple nonrecursive combinatorial formula for any kazhdan lusztig polynomial of any coxeter group theorem 4 1 and to study some consequences of it the main idea involved in the proof and statement of this formula is that of extending the concept of the rpolynomial see e g 12 w to any finite multichain of w so that for multichains of length 1 one obtains apart from sign the usual r polynomials once this has been done then the kazhdan lusztig polynomial of a pair u v turns out to be just the sum over all multichains from u to v of the corresponding generalized r polynomials the r polynomial of a multichain can be readily defined and computed from the ordinary r polynomials see 9 10 and 11 since several combinatorial formulas and interpretations are known for these polynomials see e g 5 10 and for the case of symmetric groups 3 and simple recurrences exist for them we feel that this formula is a significant step forward in the understanding of the kazhdanlusztig polynomials though combinatorial formulas for kazhdan lusztig polynomials have appeared before in the literature see e g 16 21 8 6 none of them hold in complete generality

lattice paths and kazhdan lusztig polynomials in their fundamental paper 18 kazhdan and lusztig defined for every coxeter group w a family of polynomials indexed by pairs of elements of w which have become known as the kazhdan lusztig polynomials of w see e g 17 chap 7 these polynomials are intimately related to the bruhat order of w and to the geometry of schubert varieties and have proven to be of fundamental importance in representation theory the purpose of this paper is to present a new non recursive combinatorial formula for these polynomials more precisely we show that each directed path in the bruhat graph of w has a naturally associated set of lattice paths with the property that the kazhdan lusztig polynomial of u v is the sum over all the lattice paths associated to all the paths going from u to v of l r o d r q i v i u r 1 r 2 where f o d f and f l f are three natural statistics on the lattice path we believe that this formula is the most explicit non recursive formula known for the kazhdan lusztig polynomials which holds in complete generality the organization of the paper is as follows in section 3 we define and study the r polynomial of a chain this polynomial reduces to the usual r polynomial for a chain of length one and is fundamental in all that follows in section 4 we show that the antisymmetrization of the kazhdan lusztig polynomial of two elements u v of w equals the alternating sum over all the chains going from u to v of the corresponding r polynomials in section 5 we derive some consequences of this result which although not needed in what follows are of independent interest more precisely we derive explicit formulas for each coefficient of any kazhdan lusztig polynomial and comparing these to known formulas we obtain some new identities for the r polynomials in section 6 we introduce and study a family of polynomials indexed by sequences of positive integers which is needed in the proof of our main result these polynomials are independent of w are easily computable by simple recursions and can be interpreted as counting certain lattice paths in section 7 we prove our main theorem which is obtained by combining the results in sections 4 and 6 this expresses the kazhdan lusztig polynomial of two elements u v in w as

antichains in weight posets associated with gradings of simple lie algebras for a reductive lie algebra mathfrak h and a simple finite dimensional mathfrak h module v the set of weights of v mathcal p v is equipped with a natural partial order we consider antichains in the weight poset mathcal p v and a certain operator mathfrak x acting on antichains eventually we impose stronger constraints on mathfrak h v and stick to the case in which mathfrak h mathfrak g 0 and v mathfrak g 1 for a mathbb z grading mathfrak g bigoplus i in mathbb z mathfrak g i of a simple lie algebra mathfrak g then v is a weight multiplicity free mathfrak h module and mathcal p v can be regarded as a subposet of delta where delta is the root system of mathfrak g our goal is to demonstrate that the weight posets associated with mathbb z gradings exhibit many good properties that are similar to those of delta that are observed earlier in panyushev eur j combin 30 2 586 594 2009

rings with lexicographic straightening law the concept of a straightening law as a means of analyzing the structure of particular commutative algebras has appeared independently in the work of a number of authors the purpose of this paper is to introduce a systematic theory of algebras endowed with a structure of a lexicographic straightening law based on a partially ordered set or lexicographic ring for short much of the work in the literature dealing with such rings fits quite naturally in the context of our theory and we sketch some of this work moreover some of our later results in particular the betti number bound in section 6 may actually be new results even for the special cases of the known examples of lexicographic rings the concept of a lexicographic ring as an interesting area of study was suggested to this author by deconcini who conjectured that a lexicographic ring is cohen macaulay if the underlying partially ordered set is so we prove this result in two different ways it has recently come to our attention that this conjecture has also been proved by deconcini et al 131 using deformation theory methods the results of this paper are arranged in two parts the first part consisting of sections 2 through 4 is more elementary and uses the technique of combinatorial decompositions as developed by garsia and this author the second part sections 6 and 7 requires some knowledge of homological algebra methods in ring theory the intermediate section 5 discusses some of the known examples of lexicographic rings in this section we also sketch some of the ways that our theory may be extended to more general contexts the main result of the first part is the transfer theorem 4 3 which enables one to transfer combinatorial decompositions from the underlying partially ordered set or more precisely from the corresponding stanley reisner ring to the lexicographic ring as a result those concepts that are expressible in terms of combinatorial decompositions are also

fixed points of involutive automorphisms of the bruhat order abstract applying a classical theorem of smith we show that the poset property of being gorenstein over z 2 is inherited by the subposet of fixed points under an involutive poset automorphism as an application we prove that every interval in the bruhat order on twisted involutions in an arbitrary coxeter group has this property and we find the rank function this implies results conjectured by f incitti we also show that the bruhat order on the fixed points of an involutive automorphism induced by a coxeter graph automorphism is isomorphic to the bruhat order on the fixed subgroup viewed as a coxeter group in its own right

a partial order on the regions of ⁿ dissected by hyperplanes we study a partial order on the regions of r dissected by hyperplanes this includes a computation of the mobius function and in some cases of the homotopy type applications are presented to zonotopes the weak bruhat order on weyl groups and acyclic orientations of graphs 0 introduction let cc h h2 hk be a set of hyperplanes in rw then the components of rw uheh form a set i of open n cells we will call regions traditionally i has been studied in terms of enumeration for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in c for a thorough discussion of this problem see zaslavsky s monograph zi and the references therein see also os and te in this paper we will study i from a different perspective our concern will be the way in which one can move from one region to another by walking through one hyperplane at a time to analyze this situation we define a partial order on j relative to a base region where the cover relations correspond to walking through a hyperplane and moving further away from the base the paper is organized as follows in 1 we define the partial order on rfl and begin to study its structure this culminates in theorem 1 11 where an explicit computation of its mbbius function is performed in 11 we examine the special case where all the hyperplanes go through the origin in certain cases we are able to compute the homotopy type of these posets the next three sections are devoted to applications 111 applies our work to a partial order on a zonotope iv deals with the weak bruhat order of a weyl group as a special case of our posets as a corollary we show that the homotopy type of the weak bruhat order is that of a sphere finally v applies our results to a partial order on the acyclic orientations of a graph some of the relationships used in iii iv and v appear in an enumerative context in the excellent article of greene and zaslavsky gz most of our poset notation is standard and can be found in a when more than one partial order is being used we will sometimes use subscripts to indicate where the relation is i e x py means that x is less than or equal to y in the poset p by l p we mean the simplicial complex of chains totally ordered sets in p we will think of p as its geometric realization by p we mean the order dual of p i e received by the editors july 13 1982 and in revised form may 15 1983 1980 mathematics subject classification primary 06a10 51m20 1984 american mathematical society 0002 9947 84 1 00 25 per page

sharp bounds for the first non zero stekloff eigenvalues abstract let m be an n 2 dimensional compact riemannian manifold with boundary and non negative ricci curvature consider the following two stekloff eigenvalue problems u 0 in m u p u on m 2 u 0 in m u u q u 0 on m where is the laplacian operator on m and denotes the outward unit normal on m the first non zero eigenvalues of the above problems will be denoted by p 1 and q 1 respectively in the present paper we prove that if the principle curvatures of the second fundamental form of m are bounded below by a positive constant c then p 1 1 1 1 n 1 c 2 n 1 c with equality holding if and only if is isometric to an n dimensional euclidean ball of radius 1 c here 1 denotes the first non zero eigenvalue of the laplacian of m we also show that if the mean curvature of m is bounded below by a positive constant c then q 1 n c with equality holding if and only if m is isometric to an n dimensional euclidean ball of radius 1 c finally we show that q 1 a v and that if the equality holds and if there is a point x 0 m such that the mean curvature of m at x 0 is no less than a n v then m is isometric to an n dimensional euclidean ball being a and v the area of m and the volume of m respectively

a higher dimensional generalization of hersch payne schiffer inequality for steklov eigenvalues abstract in this paper we generalize the hersch payne schiffer inequality for steklov eigenvalues to higher dimensional case by extending the trick used by hersch payne and schiffer to higher dimensional manifolds more precisely we show that for a compact oriented riemannian manifold with boundary of dimension n the multiplication of a steklov eigenvalue for functions and a steklov eigenvalue for differential n 2 forms can be controlled from above by a certain eigenvalue of the laplacian operator on the boundary

on the convergence of formal equivalence between embeddings let j x w denote the morphism embedding the compact complex manifold x into the manifold w the formal principle holds for j if given any other embedding j x w together with a formal isomorphism between j and j there is an actual isomorphism between j and j for definitions see 2 1 for instance the formal principle holds for exceptional curves of the first kind on surfaces on the other hand arnold 1 recently gave the example of an embedding with a topologically trivial normal bundle of an elliptic curve into a surface for which the formal principle fails a number of articles dealing with the formal principle in algebraic as well as in analytic geometry have been published nirenberg spencer 19 grauert 7 hironaka rossi 16 griffiths 9 10 hironaka 14 hironaka matsumura 15 hartshorne 11 artin 2 gieseker 6 moreover a few weeks after the present work was written commichaugrauert 23 proved that the formal principle holds under an assumption one positivity which seems much weaker than ours we hope that the present article retains some interest because of its method which may work in the algebraic and or singular case we are interested in the complex analytic case in the situation where the normal bundle is fairly positive or non negative this means there are enough submanifolds in w around x we propose a conjecture and then give our result we let d denote the germ of the space of submanifolds near x an open subset in the douady space ud denote the universal submanifold of d x w and lid denote the projection from ud into w

some combinatorial and algebraic properties of coxeter complexes and tits buildings a des groupes de coxeter et des groupes a bn paire sont associes certaines structures combinatoires les complexes de coxeter et les constructions de tits on etudie ces structures du point de vue des complexes de cohen macaulay et des complexes decorticables

arithmetic cohen macaulayness and arithmetic normality for schubert varieties let g be a semisimple simply connected chevalley group over a field k let t be a maximal torus b a borel subgroup d t and w the weyl group of g relative to t let p be a maximal parabolic subgroup of classical type cf 7 8 i e a maximal parabolic subgroup such that if w is the associated fundamental weight then i w av i 1 2 w c a is 2 for all roots ae being a w invariant scalar product on x t q where x t character group of t for t e w wp let x t denote the schubert variety in g p associated to t namely x t is the zariski closure btp mod p in g p of the bruhat cell endowed with the canonical reduced structure for the canonical projective embedding g p in p h g p l let rt denote the homogeneous co ordinate ring of x t then the main result of this paper is given by

on hyperboundedness and spectrum of markov operators consider an ergodic markov operator m reversible with respect to a probability measure mu on a general measurable space it is shown that if m is bounded from mathbb l 2 mu to mathbb l p mu where p 2 then it admits a spectral gap this result answers positively a conjecture raised by hoegh krohn and simon j funct anal 9 121 80 1972 in the more restricted semi group context the proof is based on isoperimetric considerations and especially on cheeger inequalities of higher order for weighted finite graphs recently obtained by lee et al proceedings of the 2012 acm symposium on theory of computing 1131 1140 acm new york 2012 it provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general in addition the usual cheeger inequality is extended to the higher eigenvalues in the compact riemannian setting and the exponential behaviors of the small eigenvalues of witten laplacians at small temperature are recovered

shellability in reductive monoids the purpose of this paper is to extend to monoids the work of bjorner wachs and proctor on the shellability of the bruhat chevalley order on weyl groups let m be a reductive monoid with unit group g borel subgroup b and weyl group w we study the partially ordered set of b x b orbits with respect to zariski closure inclusion within a g x g orbit of m this is the same as studying a w x w orbit in the renner monoid r such an orbit is the retract of a universal orbit which is shown to be lexicograhically shellable in the sense of bjorner and wachs

comparison of steklov eigenvalues on a domain and laplacian eigenvalues on its boundary in riemannian manifolds abstract we prove that in riemannian manifolds the k th steklov eigenvalue on a domain and the square root of the k th laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions as an application we derive a weyl type upper bound for steklov eigenvalues a pohozaev type identity for harmonic functions on the domain and the min max variational characterization of both eigenvalues are important ingredients

topology of posets with special partial matchings abstract special partial matchings spms are a generalisation of brenti s special matchings let a pircon be a poset in which every non trivial principal order ideal is finite and admits an spm thus pircons generalise marietti s zircons we prove that every open interval in a pircon is a pl ball or a pl sphere it is then demonstrated that bruhat orders on certain twisted identities and quasiparabolic w sets constitute pircons together these results extend a result of can cherniavsky and twelbeck prove a conjecture of hultman and confirm a claim of rains and vazirani

combinatorial proof of the inversion formula on the kazhdan lusztig r polynomials in this paper we present a combinatorial proof of the inversion formula on the kazhdan lusztig r polynomials this problem was raised by brenti as a consequence we obtain a combinatorial interpretation of the equidistribution property due to verma stating that any nontrivial interval of a coxeter group in the bruhat order has as many elements of even length as elements of odd length the same argument leads to a combinatorial proof of an extension of verma s equidistribution to the parabolic quotients of a coxeter group obtained by deodhar as another application we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element

on the lie algebroid of a derived self intersection abstract let i x y be a closed embedding of smooth algebraic varieties denote by n the normal bundle of x in y the present paper contains two constructions of certain lie structure on the shifted normal bundle n 1 encoding the information of the formal neighborhood of x in y we also present a few applications of these lie theoretic constructions in understanding the algebraic geometry of embeddings

shellability is np complete we prove that for every d 2 deciding if a pure d dimensional simplicial complex is shellable is np hard hence np complete this resolves a question raised e g by danaraj and klee in 1978 our reduction also yields that for every d 2 and k 0 deciding if a pure d dimensional simplicial complex is k decomposable is np hard for d 3 both problems remain np hard when restricted to contractible pure d dimensional complexes another simple corollary of our result is that it is np hard to decide whether a given poset is cl shellable