on pseudo inverses of matrices and their characteristic polynomials in supertropical algebra abstract the only invertible matrices in tropical algebra are diagonal matrices permutation matrices and their products however the pseudo inverse a defined as 1 det a adj a with det a being the tropical permanent also called the tropical determinant of a matrix a inherits some classical algebraic properties and has some surprising new ones defining b and b to be tropically similar if b a b a we examine the characteristic max polynomials of tropically similar matrices as well as those of pseudo inverses other miscellaneous results include a new proof of the identity for det a b and a connection to stabilization of the powers of definite matrices

tropical compound matrix identities abstract we prove identities on compound matrices in extended tropical semirings such identities include analogues to properties of conjugate matrices powers of matrices and adj a det a 1 all of which have implications on the eigenvalues of the corresponding matrices a tropical sylvester franke identity is provided as well

subdivisions and local h vectors in part i a general theory of f vectors of simplicial subdivisions or triangulations of simplicial complexes is developed based on the concept of lo cal h vector as an application we prove that the h vector of a cohen macaulay complex increases under quasi geometric subdivision thus establishing a spe cial case of a conjecture of kalai and this author techniques include com mutative algebra homological algebra and the intersection homology of toric varieties in part ii we extend the work of part i to more general situations first a formal generalization of subdivision is given based on incidence alge bras special cases are then developed in particular one based on subdivisions of eulerian posets and involving generalized h vectors other cases deal with kazhdan lusztig polynomials ehrhart polynomials and a q analogue of eule rian posets many applications and examples are given throughout department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address rstan math mit edu this content downloaded from 157 55 39 210 on sat 30 jul 2016 05 59 16 utc all use subject to http about jstor org terms

t increasing paths on the bruhat graph of affine weyl groups are self avoiding abstract let w s be a coxeter system t wsw 1 s s w w its set of reflections any total reflection order and the undirected bruhat graph we consider the natural labeling of the edges of which assigns to the edge v w the reflection vw 1 a path on i e a sequence v 1 v k such that v i v 1 i 1 t for i 1 k 1 is called t increasing if v 1 v 1 2 v k 1 v 1 k t increasing paths play an important role in the computation of both the kazhdan lusztig and the r polynomials of w we prove that if w is finite or is an affine weyl group then any t increasing path is self avoiding i e it has no self intersection points

note sur les polynomes de kazhdan lusztig in his article brenti j am math soc 11 2 480 497 1973 brenti gave a non recursive formula for the kazhdan lusztig polynomials of a coxeter group and proved it by combinatorial methods cf theorem 4 1 of brenti j am math soc 11 2 480 497 1973 the goal of this note is to give a geometric interpretation of this formula for the coxeter groups that are isomorphic to the weyl group of a split group over a finite field this interpretation rests on a result of morel j am math soc 21 1 23 61 2008 theorem 3 3 5 that expresses the intermediate extension of a pure perverse sheaf as a weight truncation of the usual direct image

a three point formula for numerical quadrature of oscillatory integrals with variable frequency abstract a simple three point formula is constructed for the evaluation of general oscillatory integrals a rigorous derivation of the local error term is presented and the implications to high frequency oscillations are discussed simple examples given include integrals with variable frequency for which the usual filon formula would be inappropriate for cases where filon s formula is appropriate the new formula appears to be computationally more efficient the main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach comparison with exact results is possible from the work of stoker 16 for a case which whilst special in the physical sense does not simplify the integral involved in all cases the implementation of the formula is as straightforward as the implementation of the ordinary simpson rule

special matchings and kazhdan lusztig polynomials abstract in 1979 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 and which have proven to be of importance in several areas of mathematics in this paper we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials our results also imply and generalize the recent one in adv in math 180 2003 146 175 on the combinatorial invariance of kazhdan lusztig polynomials

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

supertropical matrix algebra iii powers of matrices and their supertropical eigenvalues abstract we investigate powers of supertropical matrices with special attention to the role of the coefficients of the supertropical characteristic polynomial especially the supertropical trace in controlling the rank of a power of a matrix this leads to a jordan type decomposition of supertropical matrices together with a supertropical eigenspace decomposition of a power of an arbitrary supertropical matrix

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

explicit formulae for the brenti s polynomials γa1 ar abstract we introduce two kinds of generalized catalan numbers a b a b z and a1 ar h a i p h n they can be interpreted as the numbers of certain bounded lattice paths in a lattice point set n n 1 three explicit formulae for the brenti s polynomials a1 ar are given in terms of generalized catalan numbers by these formulae we study the properties of a1 ar and answer some problems of brenti

inversion of matrices over a commutative semiring it is a well known consequence of the elementary theory of vector spaces that if a and b are n by n matrices over a field or even a skew field such that ab 1 then ba 1 this result remains true for matrices over a commutative ring however it is not in general true for matrices over noncommutatives rings in this paper we show that if a and b are n by n matrices over a commutative semiring then the equation ab 1 implies ba 1 we give two proofs one algebraic in nature the other more combinatorial both proofs use a generalization of the familiar product law for determinants over a commutative semiring

on the error and its control in a two parameter generalised newton cotes rule a quadrature formula containing two free phase parameters k k and recently written by van daele et al 13 is rederived using an extension of lagrange s identity by using this method a closed form expression is determined for the local error term e f and the relevant peano kernel is given explicitly sufficient conditions are established under which this kernel remains definite thus allowing a particularly simple expression to be deduced for e f which reduces to the classical result for boole s rule in the limit k k 0 recent work on optimisation of global error is extended to include the new rule it is shown that this error can be reduced by a factor o h 2 on a certain curve in the phase space a further reduction by a factor o h 2 is sometimes possible by choosing the phase parameters on the intersection of this curve and another such when these curves intersect only at complex values the reduction is still achieved a closed form expression for the global error is also derived under these circumstances and this is seen to be asymptotically o h 1 0 as h 0 a second limiting form of the five point formula is found to reduce to the generalised boole s rule written by vanden berghe et al 14 and a third limiting form is also written this third rule which is a special case of a one parameter family of generalised boole s rules is seen to perform better than the other two in the two examples studied several numerical examples are given with extensive diagrams to illustrate all uses of the techniques proposed

p kernels ic bases and kazhdan lusztig polynomials abstract in j amer math soc 5 1992 805 851 stanley introduced the concept of a p kernel for any locally finite partially ordered set p in proc sympos pure math vol 56 ams 1994 pp 135 148 du introduced for any set p the concept of an ic basis the purpose of this article is to show that under some mild hypotheses these two concepts are equivalent and to characterize for a given coxeter group w partially ordered by bruhat order the w kernel corresponding to the kazhdan lusztig basis of the hecke algebra of w finally we show that this w kernel factorizes as a product of other w kernels and that these provide a solution to the yang baxter equations for w

rigidity of schubert closures and invariance of kazhdan lusztig polynomials abstract let e w e w be intervals from the origin in two coxeter groups w w we prove that any poset isomorphism e w e w preserves kazhdan lusztig polynomials in the sense that p x y px y for any x y in e w in the case where one of the two groups w w has the property that the coxeter graph of each of its irreducible constituents is either a tree or affine of type a n in particular the result holds for all finite or affine coxeter groups we obtain this result as a consequence of an explicit description of all the poset isomorphisms from e w to e w

on the characteristic polynomial of a supertropical adjoint matrix abstract let a denote the characteristic polynomial of a matrix a over a field a standard result of linear algebra states that a 1 is the reciprocal polynomial of a more formally the condition n a k a 1 n k a holds for any invertible n n matrix a over a field where i a denotes the coefficient of n i in the characteristic polynomial det i a we confirm a recent conjecture of niv by proving the tropical analogue of this result

from moment graphs to intersection cohomology we describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori in terms of structure of the zero and one dimensional orbits the class of varieties to which our formula applies includes schubert varieties in flag varieties and affine flag varieties we also prove a monotonicity result on local intersection cohomology stalks

lattice paths lexicographic correspondence and kazhdan lusztig polynomials abstract in this paper we give a new closed formula for the kazhdan lusztig polynomials of finite coxeter groups and affine weyl groups this formula is computationally more efficient than any existing one and it conjecturally holds for all coxeter groups

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

the structure of total reflection orders in affine root systems abstract we analyze the structure of total reflection orders in affine root systems we obtain a complete characterization of the finite sections of these orders

boolean elements in kazhdan lusztig theory abstract kazhdan lusztig polynomials have been proven to play an important role in different fields despite this there are still few explicit formulae for them here we give closed product formulae for the kazhdan lusztig polynomials indexed by boolean elements in a class of coxeter systems that we call linear boolean elements are elements smaller than a reflection that admits a reduced expression of the form s 1 s n 1 s n s n 1 s 1 s i s s i s j if i j then we provide several applications of this result concerning the combinatorial invariance of the kazhdan lusztig polynomials the classification of the pairs u v with u v the kazhdan lusztig elements and the intersection homology poincare polynomials of the schubert varieties

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

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

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

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

a note on the computation of a green s function for short wavelength water waves over a plane incline a recently developed green s function for two dimensional water wave problems on a plane beach was given in a form requiring in part direct numerical quadrature of a complex valued integral on the bromwich line s 12 s these calculations would fail numerically for waves of sufficiently short wave length for special angles of inclination a more robust method is presented here which has no wavelength restrictions

a recursive rule for kazhdan lusztig characters abstract a formula for computing kazhdan lusztig characters in hecke algebras of coxeter groups is given this formula holds for products of coxeter elements of parabolic subgroups in an arbitrary coxeter group where the parabolic subgroups are of type a and the kazhdan lusztig representations are of finite dimension the formula yields new combinatorial interpretations of murnaghan nakayama type to these characters and to other related characters

euler flag enumeration of whitney stratified spaces abstract the flag vector contains all the face incidence data of a polytope and in the poset setting the chain enumerative data it is a classical result due to bayer and klapper that for face lattices of polytopes and more generally eulerian graded posets the flag vector can be written as a c d index a non commutative polynomial which removes all the linear redundancies among the flag vector entries this result holds for regular cw complexes we relax the regularity condition to show the c d index exists for whitney stratified manifolds by extending the notion of a graded poset to that of a quasi graded poset this is a poset endowed with an order preserving rank function and a weighted zeta function this allows us to generalize the classical notion of eulerianness and obtain a c d index in the quasi graded poset arena we also extend the semi suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

peak algebras paths in the bruhat graph and kazhdan lusztig polynomials we obtain a nonrecursive combinatorial formula for the kazhdan lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one and which cannot be linearly simplified our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

homotopy type of the boolean complex of a coxeter system in any coxeter group the set of elements whose principal order ideals are boolean forms a simplicial poset under the bruhat order this simplicial poset defines a cell complex called the boolean complex in this paper it is shown that for any coxeter system of rank n the boolean complex is homotopy equivalent to a wedge of n 1 dimensional spheres the number of such spheres can be computed recursively from the unlabeled coxeter graph and defines a new graph invariant called the boolean number specific calculations of the boolean number are given for all finite and affine irreducible coxeter systems as well as for systems with graphs that are disconnected complete or stars one implication of these results is that the boolean complex is contractible if and only if a generator of the coxeter system is in the center of the group

minimal max realization of convex sequences we show that the minimal dimension of a linear realization over the max semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps the minimal dimensional realization of any convex realizable sequence can thus be found in linear time the result is based on a bound in terms of minors of the hankel matrix

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra