a polynomial expression for the hilbert series of the quotient ring of diagonal coinvariants a special case of haiman s identity invent math 149 2002 pp 371 407 for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded hilbert series as a sum of rational functions in q t in this paper we show how a summation identity of garsia and zabrocki for macdonald polynomial pieri coefficients can be used to transform haiman s formula for the hilbert series into an explicit polynomial in q t with integer coefficients we also provide an equivalent formula for the hilbert series as the constant term in a multivariate laurent series

de rham complexes of q analog of general linear group glq n in this paper a set of de rham complexes of quantum group glq n determined by one parameter r is given and it is proven that the differential calculi on the quantum group glq n given in this paper are bicovariant the noncommutative differential calculi on the quantum groups slq n and suq n are also discussed

relativistic and newtonian κ space times the deformations of the galilei algebra and their associated noncommutative newtonian space times are investigated this is done by analyzing the possible nonrelativistic limits of an eleven generator pseudo extended poincare algebra p and their implications for the existence of a first order differential calculus the additional one form needed to achieve a consistent calculus on minkowski space is shown to be related to the additional central generator entering in the p hopf algebra in the process deformations of the extended galilei and galilei algebras are introduced which have respectively a cocycle and a bicrossproduct structure

classification of bicovariant differential calculi on quantum groups suppose thatq is not a root of unity we classify all bicovariant differential calculi of dimension greater than one on the quantum groupsgl q n o q n andsp q n for which the differentials du of the matrix entriesu generate the left module of first order forms our first classification theorem asserts that there are precisely two one parameter families of such calculi ongl q n forn 3 in the limitq 1 only two of these calculi give the ordinary differential calculus ongl n our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties ono q n andsp q n forn 4 this strengthens the corresponding result proved in our previous paper ss2 there are four such calculi ono q 3 we introduce two new 4 dimensional bicovariant differential calculi ono q 3

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

computational homogenization of microfractured continua using weakly periodic boundary conditions computational homogenization of elastic media with stationary cracks is considered whereby the macroscale stress is obtained by solving a boundary value problem on a statistical volume element sve and the cracks are represented by means of the extended finite element method xfem with the presence of cracks on the microscale conventional bcs dirichlet neumann strong periodic perform poorly in particular when cracks intersect the sve boundary as a remedy we herein propose to use a mixed variational format to impose periodic boundary conditions in a weak sense on the sve within this framework we develop a novel traction approximation that is suitable when cracks intersect the sve boundary our main result is the proposition of a stable traction approximation that is piecewise constant between crack boundary intersections in particular we prove analytically that the proposed approximation is stable in terms of the lbb inf sup condition and illustrate the stability properties with a numerical example we emphasize that the stability analysis is carried out within the setting of weakly periodic boundary conditions but it also applies to other mixed problems with similar structure e g contact problems the numerical examples show that the proposed traction approximation is more efficient than conventional boundary conditions dirichlet neumann strong periodic in terms of convergence with increasing sve size

a positivity result in the theory of macdonald polynomials we outline here a proof that a certain rational function c n q t which has come to be known as the q t catalan is in fact a polynomial with positive integer coefficients this has been an open problem since 1994 because c n q t evaluates to the catalan number at t q 1 it has also been an open problem to find a pair of statistics a b on the collection n of dyck paths pi of length 2n yielding c n q t summation operator pi t a pi q b pi our proof is based on a recursion for c n q t suggested by a pair of statistics recently proposed by j haglund one of the byproducts of our results is a proof of the validity of haglund s conjecture

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

classification of differential calculi on uq b classical limits and duality we give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra uq b which we view as the quantum function algebra cq b here b is the borel subalgebra of sl2 we do the same in the classical limit q 1 and obtain a one to one correspondence in the finite dimensional case it turns out that the classification is essentially given by finite subsets of the positive integers we proceed to investigate the classical limit from the dual point of view i e with function algebra u b and enveloping algebra c b in this case there are many more differential calculi than coming from the q deformed setting as an application we give the natural intrinsic four dimensional calculus of minkowski space and the associated formal integral

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

on poisson lie structure on the external algebra of the classical lie groups the general expression for the bicovariant bracket for odd generators of the external algebra on a poisson lie group is given the graded poisson lie structures derived before for gl n and sl n are shown to be the special cases of this bracket the formula is universal and can be applied to any matrix lie group

on the noncommutative riemannian geometry of glq n a recently proposed definition of a linear connection in noncommutative geometry based on a generalized permutation is used to construct linear connections on glq n restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed candidates for generalized permutations on glq n are found it is shown that for a given generalized permutation there exists one and only one associated linear connection properties of the linear connection are discussed in particular its bicovariance torsion and commutative limit

quantum deformation of brst algebra we investigate theq deformation of the brst algebra the algebra of the ghost matter and gauge fields on one spacetime point using the result of the bicovariant differential caculus there are two nilpotent operations in the algebra the brst transformation b and the derivatived we show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operations as well as the operation the antimultiplicativ e inner involution

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

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

two scale finite element modelling of reinforced concrete structures effective response and subscale fracture development a two scale model is derived from a fully resolved model where the response of concrete steel reinforcement and bond between them are considered the pertinent effective large scale problem is derived from selective homogenisation in terms of the equilibrium of reinforced concrete considered as a single phase solid variational formulations of the representative volume element problem are established in terms of the subscale displacement fields for the plain concrete continuum and the reinforcement bars dirichlet and neumann boundary conditions bcs are imposed on the concrete pertaining to uniform boundary displacement and constant boundary traction respectively and on the reinforcement bars pertaining to prescribed boundary displacement and vanishing sectional forces respectively different representative volume element sizes and combinations of bcs were used in fe 2 analyses of a deep beam subjected to four point bending results were compared with those of full resolution single scale the most reliable response was obtained for the case of dirichlet dirichlet bcs with a good match between the models in terms of the deformed shape force deflection relation and average strain even though the maximum crack widths were underestimated the dirichlet dirichlet combination provided an approximate upper bound on the structural stiffness

bicovariant differential calculi on glp q 2 and quantum subgroups in a previous paper all bicovariant differential calculi on the two parameter quantum group glp q 2 were determined in this work the authors elaborate some of their properties and discuss relations with work by other authors furthermore they show that there are two different ways in which bicovariant differential calculi on glq 2 where p q induce corresponding calculi on the quantum subgroup slq 2 it is shown that on slq 2 there are only two different bicovariant differential calculi from these one obtains the 4d or calculi on suq 2 the classical limit of the two differential calculi on slq 2 is investigated revealing a relation with recent work on deformed differential calculi on commutative function algebras

ordered set partitions generalized coinvariant algebras and the delta conjecture abstract the symmetric group s n acts on the polynomial ring q x n q x 1 x n by variable permutation the invariant ideal i n is the ideal generated by all s n invariant polynomials with vanishing constant term the quotient r n q x n i n is called the coinvariant algebra the coinvariant algebra r n has received a great deal of study in algebraic and geometric combinatorics we introduce a generalization i n k q x n of the ideal i n indexed by two positive integers k n the corresponding quotient r n k q x n i n k carries a graded action of s n and specializes to r n when k n we generalize many of the nice properties of r n to r n k in particular we describe the hilbert series of r n k give extensions of the artin and garsia stanton monomial bases of r n to r n k determine the reduced grobner basis for i n k with respect to the lexicographic monomial order and describe the graded frobenius series of r n k just as the combinatorics of r n are controlled by permutations in s n we will show that the combinatorics of r n k are controlled by ordered set partitions of 1 2 n with k blocks the delta conjecture of haglund remmel and wilson is a generalization of the shuffle conjecture in the theory of diagonal coinvariants we will show that the graded frobenius series of r n k is up to a minor twist the t 0 specialization of the combinatorial side of the delta conjecture it remains an open problem to give a bigraded s n module v n k whose frobenius image is even conjecturally equal to any of the expressions in the delta conjecture our module r n k solves this problem in the specialization t 0

homogenization based multiscale crack modelling from micro diffusive damage to macro cracks the existence of a representative volume element rve for a class of quasi brittle materials having a random heterogeneous microstructure in tensile shear and mixed mode loading is demonstrated by deriving traction separation relations which are objective with respect to rve size a computational homogenization based multiscale crack modelling framework implemented in an fe2 setting for quasi brittle solids with complex random microstructure is presented the objectivity of the macroscopic response to the micro sample size is shown by numerical simulations therefore a homogenization scheme which is objective with respect to macroscopic discretization and microscopic sample size is devised numerical examples including a comparison with direct numerical simulation are given to demonstrate the performance of the proposed method

yang baxter matrix and calculi on quantum groups of a series the authors prove that the differential calculi on quantum groups of a series given in this letter and in previous papers are calculi

the quantized universal enveloping algebras uq iso n uq e 3 1 and uq e n and the representation theory of uq e 3 we construct the hopf algebras of uq iso n and uq iso 3 1 as regular functionals on their function algebras examine the possible involutions on these objects derive the quantized euclidean algebras uq e n and finally give the irreducible representations of uq e 3 depending on two parameters which fix the value of the casimir operators

haglund s conjecture on 3 column macdonald polynomials we prove a positive combinatorial formula for the schur expansion of llt polynomials indexed by a 3 tuple of skew shapes this verifies a conjecture of haglund proc natl acad sci usa 101 46 16127 16131 2004 the proof requires expressing a noncommutative schur function as a positive sum of monomials in lam s eur j combin 29 1 343 359 2008 algebra of ribbon schur operators combining this result with the expression of haglund et al j am math soc 18 3 735 761 2005 for transformed macdonald polynomials in terms of llt polynomials then yields a positive combinatorial rule for transformed macdonald polynomials indexed by a shape with 3 columns

combinatorial theory of macdonald polynomials i proof of haglund s formula abstract haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h we give a combinatorial proof of this conjecture which establishes the existence and integrality of h as corollaries we obtain the cocharge formula of lascoux and schutzenberger for hall littlewood polynomials a formula of sahi and knop for jack s symmetric functions a generalization of this result to the integral macdonald polynomials j a formula for h in terms of lascoux leclerc thibon polynomials and combinatorial expressions for the kostka macdonald coefficients k when is a two column shape

conjectured statistics for the q t catalan numbers abstract we introduce the distribution function f n q t of a pair of statistics on catalan words and conjecture f n q t equals garsia and haiman s q t catalan sequence c n q t which they defined as a sum of rational functions we show that f n s q t defined as the sum of these statistics restricted to catalan words ending in s ones satisfies a recurrence relation as a corollary we are able to verify that f n q t c n q t when t 1 q we also show the partial symmetry relation f n q 1 f n 1 q by modifying a proof of haiman of a q lagrange inversion formula based on results of garsia and gessel we obtain a q analogue of the general lagrange inversion formula which involves catalan words grouped according to the number of ones at the end of the word

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

differential hopf algebra structure of the quantum standard complex we are investigating the quantum standard complex k q g d of the quantum enveloping algebra uq g for lie algebras g associated with the root systems an bn cn and dn it is a quantum version of the standard koszul complex associated to a lie algebra as applied for instance in the brs quantization procedure in connection with spin representations using techniques from the theory of braided monoidal categories we obtain a differential hopf algebra structure on the complex k q g d

on fock space representations of quantized enveloping algebras related to noncommutative differential geometry in this paper explicitly natural from the geometrical point of view fock space representations contragradient verma modules of the quantized enveloping algebras are constructed in order to do so one starts from the gauss decomposition of the quantum group and introduces the differential operators on the corresponding q deformed flag manifold assumed as a left comodule for the quantum group by a projection to it of the right action of the quantized enveloping algebra on the quantum group finally the representatives of the elements of the quantized enveloping algebra corresponding to the left invariant vector fields on the quantum group are expressed as first order differential operators on the q deformed flag manifold

bicovariant 3d calculus for slq 2 from poisson lie structures all possible graded poisson lie structures on the external algebra of sl 2 are described we prove that differential poisson lie structures prolonging the sklyanin brackets do not exist on sl 2 there are two and only two graded poisson lie structures on sl 2 and neither of them can be obtained by a reduction of graded poisson lie structures on the external algebra of gl 2 both of them can be quantized and as a result we get new graded algebras of quantum right invariant forms on slq 2 with three generators

ribbon tile invariants let t be a finite set of tiles and b a set of regions f tileable by t we introduce a tile counting group g t b as a group of all linear relations for the number of times each tile t c t can occur in a tiling of a region r c b we compute the tile counting group for a large set of ribbon tiles also known as rim hooks in a context of representation theory of the symmetric group the tile counting group is presented by its set of generators which consists of certain new tile invariants in a special case these invariants generalize the conway lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group the heart of the proof is the known bijection between rim hook tableaux and certain standard skew young tableaux we also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem

the mathbb a _ q t algebra and parabolic flag hilbert schemes the earlier work of the first and the third name authors introduced the algebra mathbb a q t and its polynomial representation in this paper we construct an action of this algebra on the equivariant k theory of certain smooth strata in the flag hilbert scheme of points on the plane in this presentation the fixed points of the torus action correspond to generalized macdonald polynomials and the matrix elements of the operators have an explicit presentation

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

all bicovariant differential calculuses on glq 3 c and slq 3 c all bicovariant first order differential calculuses on the quantum group glq 3 c are determined there are two distinct one parameter families of calculuses in terms of a suitable basis of 1 forms the commutation relations can be expressed with the help of the r matrix of glq 3 c some calculuses induce bicovariant differential calculuses on slq 3 c and on real forms of glq 3 c for the generic deformation parameter q there are six calculuses on slq 3 c on suq 3 there are only two the classical limit q to 1 of bicovariant calculuses on slq 3 c is not the ordinary calculus on sl 3 c one obtains a deformation of it which involves the cartan killing metric

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

the standard complex of quantum enveloping algebras we consider quantized derivation representations of hopf algebras in some associative algebras the algebra of quantized differential operators of bialgebra representations which are special representations by derivations are constructed for the quantum enveloping algebras of lie algebras associated with the root systems and we define a deformation of the exterior algebra of forms and by applying the above results it is shown that the quantized adjoint representation of induces a bialgebra representation of in the resulting algebra of differential operators is a deformation of the standard koszul complex of lie algebras it admits an exterior derivative which is in particular a module morphism hence cohomology groups of relative to some module can be constructed

multivariate diagonal coinvariant spaces for complex reflection groups abstract for finite complex reflection groups we consider the graded w modules of diagonally harmonic polynomials in r sets of variables and show that associated hilbert series may be described in a global manner independent of the value of r

fiw modules and stability criteria for representations of classical weyl groups abstract in this paper we develop machinery for studying sequences of representations of any of the three families of classical weyl groups extending work of church ellenberg farb and nagpal 7 8 on the symmetric groups s n to the signed permutation groups b n and the even signed permutation groups d n for each family w n we present an algebraic framework where a sequence v n of w n representations is encoded into a single object we call an fi w module we prove that if an fi w module v satisfies a simple finite generation condition then the structure of the sequence is highly constrained one consequence is that the sequence is uniformly representation stable in the sense of church farb that is the pattern of irreducible representations in the decomposition of each v n eventually stabilizes in a precise sense using the theory developed here we obtain new results about the cohomology of generalized flag varieties associated to the classical weyl groups and more generally the r diagonal coinvariant algebras we analyze the algebraic structure of the category of fi w modules and introduce restriction and induction operations that enable us to study interactions between the three families of groups we use this theory to prove analogues of murnaghan s 1938 stability theorem for kronecker coefficients for the families b n and d n the theory of fi w modules gives a conceptual framework for stability results such as these

affine permutations and rational slope parking functions we introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics and relate the combinatorics of parking functi ons to that of affine permutations we relate our construction to two previously known combinatorial constructions haglund s bijectionexchanging the pairs of statistics area dinv and bounce area on dyck paths and the pak stanley labeling of the regions of k shi hyperplane arrangements by k parking functions essentially our approach can be viewed as a generalization and a unification of these two constructions we also relate o ur combinatorial constructions to representation theory we derive new formulas for the poincare polynomials of certain affine springer fibers and describe a connection to the theory of finite dimensional representations of daha and non symmetric macdonald polynomials

variationally consistent computational homogenization of transient heat flow a framework for variationally consistent homogenization combined with a generalized macrohomogeneity condition is exploited for the analysis of non linear transient heat conduction within this framework the classical approach of model based first order homogenization for stationary problems is extended to transient problems homogenization is then carried out in the spatial domain on representative volume elements rve which are in practice introduced in quadrature points in standard fashion along with the classical averages a higher order conservation quantity is obtained an iterative fe2 algorithm is devised for the case of non linear diffusion and storage coefficients and it is applied to transient heat conduction in a strongly heterogeneous particle composite parametric studies are carried out in particular with respect to the influence of the internal length associated with the second order conservation quantity copyright c 2009 john wiley sons ltd

on a volume averaged measure of macroscopic reinforcement slip in two scale modeling of reinforced concrete a two scale model for reinforced concrete in which the large scale problem formulation is enriched by an effective reinforcement slip variable is derived from the single scale model describing the response of plain concrete reinforcement steel as well as the bond between them the subscale problem on the representative volume element rve is correspondingly defined as finding the response of the rve subjected to effective variables strain slip and slip gradient imposed from the large scale a novel volumetric definition of effective reinforcement slip and its gradient is devised and the corresponding subscale problem is formulated the newly defined effective variables are imposed on the rve in a weak sense via lagrange multipliers the response of the rves of different sizes was investigated by means of pull through tests and the novel boundary condition type was used in fe 2 analyses of a deep beam locally prescribing the macroscopic reinforcement slip and its gradient in the proposed manner resulted in reduced rve size dependency of effective work conjugates which allows for more objective description of reinforcement slip in two scale modelling of reinforced concrete globally this formulation produced more consistent amplitudes of effective slip fluctuations as well as more consistent maximum crack width predictions

generalized lie algebras of type an it is shown that the quantized enveloping algebra of sl n contains a generalized lie algebra defined by means of axioms similar to woronowicz s this gives rise to lie algebra like generators and relations for the locally finite part of the quantized enveloping algebra and suggests a canonical poincare birkhoff witt basis

differential hopf algebras on quantum groups of type a abstract let a be a hopf algebra and be a bicovariant first order differential calculus over a it is known that there are three possibilities to construct a differential hopf algebra jthat contains as its first order part corresponding to the three choices of the idealj we distinguish the universal exterior algebra the second antisymmetrizer exterior algebra and woronowicz external algebra respectively let be one of then2 dimensional bicovariant first order differential calculi on the quantum groupglq n orslq n and letqbe a transcendental complex number for woronowicz external algebra we determine the dimension of the space of left invariant and of bi invariantk forms respectively bi invariant forms are closed and represent different de rham cohomology classes the algebra of bi invariant forms is graded anti commutative forn 3 the three differential hopf algebras coincide however in case of the 4d calculi onslq 2 the universal differential hopf algebra is strictly larger than woronowicz external algebra the bi invariant 1 form is not closed

square boldsymbol q t lattice paths and boldsymbol nabla p_n the combinatorial q t catalan numbers are weighted sums of dyck paths introduced by j haglund and studied extensively by haglund haiman garsia loehr and others the q t catalan numbers besides having many subtle combinatorial properties are intimately connected to symmetric functions algebraic geometry and macdonald polynomials in particular the n th q t catalan number is the hilbert series for the module of diagonal harmonic alternants in 2n variables it is also the coefficient of s 1 n in the schur expansion of e n using q t analogues of labelled dyck paths haglund et al have proposed combinatorial conjectures for the monomial expansion of e n and the hilbert series of the diagonal harmonics modules this article extends the combinatorial constructions of haglund et al to the case of lattice paths contained in squares we define and study several q t analogues of these lattice paths proving combinatorial facts that closely parallel corresponding results for the q t catalan polynomials we also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator in particular we conjecture combinatorial formulas for the monomial expansion of p n the hilbert series p n h 1 n and the sign character p n s 1 n

affine springer fibers of type a and combinatorics of diagonal coinvariants abstract we calculate the borel moore homology of affine springer fibers of type a associated with some regular semisimple nil elliptic elements as a result we obtain bigraded s n modules whose bigraded frobenius series are a generalization of the symmetric functions introduced by haglund haiman loehr remmel and ulyanov

complex quantum group dual algebra and bicovariant differential calculus the method used to construct the bicovariant bimodule in ref csww is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group the complex quantum group fun q sl n c is defined by requiring that it contains fun q su n as a subalgebra analogously to the quantum lorentz group analyzing the properties of the fundamental bimodule we show that the dual algebra has the structure of the twisted product fun q su n fun q su n reg then the bicovariant differential calculi on the complex quantum group are constructed

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

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

the representation of the symmetric group on m tamari intervals an m ballot path of size n is a path on the square grid consisting of north and east unit steps starting at 0 0 ending at mn n and never going below the line fx myg the set of these paths can be equipped with a lattice structure called them tamari lattice and denoted by t m n which generalizes the usual tamari lattice tn obtained when m 1 this lattice was introduced by f bergeron in connection with the study of diagonally coinvariant spaces in three sets ofn variables the representation of sn on these spaces is conjectured to be closely related to the natural representation of sn on labelled intervals of them tamari lattice studied in this paper an interval p q of t m n is labelled if the north steps of q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps the symmetric group sn acts on labelled intervals of t m n by permutation of the labels we prove an explicit formula conjectured by f bergeron and the third author for the character of the associated representation of sn in particular the dimension of the representation that is the number of labelledm tamari intervals of sizen is found to be m 1 n mn 1 n 2 these results are new even whenm 1 the form of these numbers suggests a connection with parking functions but our proof is not bijective the starting point is a recursive description of m tamari intervals it yields an equation for an associated generating function which is a refined version of the frobenius series of the representation the form of this equation is highly non standard it involves two additional variables x and y a derivative with respect to y and iterated divided differences with respect tox the hardest part of the proof consists in solving it and we develop original techniques to do so r

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

the delta conjecture we conjecture two combinatorial interpretations for the symmetric function eken where f is an eigenoperator for the modified macdonald polynomials defined by bergeron garsia haiman and tesler both interpretations can be seen as generalizations of the shuffle conjecture a statement originally conjectured by haglund haiman remmel loehr and ulyanov and recently proved by carlsson and mellit we show how previous work of the second and third authors on tesler matrices and ordered set partitions can be used to verify several cases of our conjectures furthermore we use a reciprocity identity and llt polynomials to prove another case finally we show how our conjectures inspire 4 variable generalizations of the catalan numbers extending work of garsia haiman and the first author

lie algebraic structure from inhomogeneous hopf algebras we construct the vector space dual to the space of right invariant differential forms constructed from a first order differential calculus on inhomogeneous quantum groups we show that this vector space is equipped with a structure of a hopf algebra which closes on a noncommutative lie algebra satisfying a jacobi identity

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

differential calculus onisoq n quantum poincare algebra and q gravity we present a general method to deform the inhomogeneous algebras of thebn cn dn type and find the corresponding bicovariant differential calculus the method is based on a projection frombn 1 cn 1 dn 1 for example we obtain the bicovariant inhomogeneousq algebraisoq n as a consistent projection of the bicovariant q algebrasoq n 2 this projection works for particular multiparametric deformations ofso n 2 the so called minimal deformations the case ofisoq 4 is studied in detail a real form corresponding to a lorentz signature exists only for one of the minimal deformations depending on one parameterq the quantum poincare lie algebra is given explicitly it has 10 generators no dilatations and contains theclassical lorentz algebra only the commutation relations involving the momenta depend onq finally we discuss aq deformation of gravity based on the gauging of thisq poincare algebra the lagrangian generalizes the usual einstein cartan lagrangian

on the algebraic structure of differential calculus on quantum groups intrinsic hopf algebra structure of the woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups coordinate functions differential 1 forms lie derivatives and inner derivations as the cross product algebra of two mutually dual graded hopf algebras this construction properly taking into account hopf algebraic properties of woronowicz s bicovariant calculus provides a direct proof of the cartan identity and of many other useful relations a detailed comparison with other approaches is also given

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

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