birational rigidity of complete intersections we prove that every smooth complete intersection x x d 1 ldots d s subset mathbb p sum i 1 s d i defined by s hypersurfaces of degree d 1 ldots d s is birationally superrigid if 5s 1 le frac 2 sum i 1 s d i 1 sqrt prod i 1 s d i in particular x is non rational and mathrm bir x mathrm aut x we also prove birational superrigidity of singular complete intersections with similar numerical condition these extend the results proved by tommaso de fernex

mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

symmetric coinvariant algebras and local weyl modules at a double point abstract the symmetric coinvariant algebra c x 1 x n s n is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin it is known that the algebra is isomorphic to the regular representation of s n replacing c x with a c x y x y we introduce another symmetric coinvariant algebra a s n n and determine its s n module structure as an application we determine the sl r 1 module structure of the local weyl module at a double point for sl r 1 a

minimize the use of minimization with unequal allocation summary minimization as an alternative to randomization is gaining popularity for small clinical trials in response to critics questions about the proper analysis of such a trial proponents have argued that a rerandomization approach akin to a permutation test with conventional randomization can be used however they add that this computationally intensive approach is not necessary because its results are very similar to those of a t test or test of proportions unless the sample size is very small we show that minimization applied with unequal allocation causes problems that challenge this conventional wisdom

garsia haiman modules for hook partitions and green polynomials with two variables abstract we consider garsia haiman modules for the symmetric group a doubly graded generalization of springer modules our main interest lies in singly graded submodules of a garsia haiman module we show that these submodules satisfy a certain combinatorial property and verify that this property is implied by a behavior of macdonald polynomials at roots of unity

macdonald positivity via the harish chandra d module using the harish chandra d module we give a proof of haiman s theorem on the positivity of macdonald polynomials ginzburg s work on the connection between this d module and the isospectral commuting variety is fundamental to this approach

asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials a general doubly adaptive biased coin design is proposed for the allocation of subjects to k treatments in a clinical trial this design follows the same spirit as efron s biased coin design and applies to the cases where the desired allocation proportions are unknown but estimated sequentially strong consistency a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions for two treatments a new family of designs is proposed and shown to be less variable than both the randomized play the winner rule and the adaptive randomized design also the proposed design tends toward a randomization scheme with a fixed target proportion as the size of the experiment increases

quantum cohomology of hilbn c2 and the weighted hook walk on young diagrams abstract following the work of okounkov and pandharipande 2010 op1 op2 diaconescu d and the recent work of i ciocan fontanine et al in preparation cdkm studying the equivariant quantum cohomology q h c 2 hilb n of the hilbert scheme and the relative donaldson thomas theory of p 1 c 2 we establish a connection between the j function of the hilbert scheme and a certain combinatorial identity in two variables this identity is then generalized to a multivariate identity which simultaneously generalizes the branching rule for the dimension of irreducible representations of the symmetric group in the staircase shape we then establish this identity by a weighted generalization of the greene nijenhuis wilf hook walk which is of independent interest

towards the multiplicative behavior of the k theoretical mckay correspondence when the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the hilbert scheme of regular orbits of nakamura then there is a natural isomorphism between the grothendieck group of this resolution and the representation ring of the group given by the bridgeland king reid map however this isomorphism is not compatible with the ring structures for the hilbert scheme of points on the affine plane we study the multiplicative behavior of this map

some complete intersection symplectic quotients in positive characteristic invariants of a vector and a covector abstract given a linear action of a group g on a k vector space v we consider the invariant ring k v v g where v is the dual space we are particularly interested in the case where v f q n and g is the group u n of all upper unipotent matrices or the group b n of all upper triangular matrices in gl n f q in fact we determine f q v v g for g u n and g b n the result is a complete intersection for all values of n and q we present explicit lists of generating invariants and their relations this makes an addition to the rather short list of doubly parametrized series of group actions whose invariant rings are known to have a uniform description

the auslander gorenstein property for z algebras abstract we provide a framework for part of the homological theory of z algebras and their generalisations directed towards analogues of the auslander gorenstein condition and the associated double ext spectral sequence that are useful for enveloping algebras of lie algebras and related rings as an application we prove the equidimensionality of the characteristic variety of an irreducible representation of the z algebra and for related representations over quantum symplectic resolutions in the special case of cherednik algebras of type a this answers a question raised by the authors

microlocalization of rational cherednik algebras we construct a microlocalization of the rational cherednik algebras h of type sn this is achieved by a quantization of the hilbert scheme hilb c2 of n points in c2 we then prove the equivalence of the category of h modules and that of modules over its microlocalization under certain conditions on the parameter

quantum and affine schubert calculus and macdonald polynomials abstract we definitively establish that the theory of symmetric macdonald polynomials aligns with quantum and affine schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong bruhat order poset on the type a affine weyl group we construct two one parameter families of functions that respectively transition positively with hall littlewood and macdonald s p functions and specialize to the representatives for schubert classes of homology and cohomology of the affine grassmannian our approach leads us to conjecture that all elements in a defining set of 3 point genus 0 gromov witten invariants for flag manifolds can be formulated as strong covers

on the mckay correspondences for the hilbert scheme of points on the affine plane the quotient of a finite dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety under appropriate assumptions the mckay correspondence relates the geometry of nice resolutions of singularities and the representations of the group for the hilbert scheme of points on the affine plane we study how different correspondences mckay dual mckay and multiplicative mckay are related to each other

the combinatorics of c fixed points in generalized calogero moser spaces and hilbert schemes abstract in this paper we study the combinatorial consequences of the relationship between rational cherednik algebras of type g l 1 n cyclic quiver varieties and hilbert schemes we classify and explicitly construct c fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles furthermore we give a combinatorial description of the bijections between c fixed points induced by the etingof ginzburg isomorphism and nakajima reflection functors we apply our results to obtain a new proof as well as a generalization of the q hook formula

birational automorphisms of fano hypersurfaces v v y v 0 between smooth three dimensional quartics v v4 v 0 v 0 4 in p is a projective isomorphism this is the original wording of their theorem in fact the arguments of im give a much stronger result namely they actually prove that smooth quartics are birationally superrigid the concept of birational rigidity is not visual being rather technical it makes little impression it is immediate implications of the property of being birationally super rigid that are really impressive for a projective variety x smooth in codimension 1 and a linear system jdj free from xed components we de ne the threshold of canonical adjunction or brie y just threshold of the pair x d setting

approaches to expanding the two arm biased coin randomization to unequal allocation while preserving the unconditional allocation ratio the paper discusses three methods for expanding the biased coin randomization bcr to unequal allocation while preserving the unconditional allocation ratio at every step the first method originally proposed in the contexts of bcr and minimization is based on mapping from an equal allocation multi arm bcr despite the improvement proposed in this paper to ensure tighter adherence to the targeted unequal allocation this method still distributes the probability mass at least as wide as the permuted block randomization pbr this works for smaller block sizes but for larger block sizes a tighter control of the imbalance in the treatment assignments is desired the second method which has two versions allows to tighten the distribution of the imbalance compared with that achieved with the pbr however the distribution of the imbalance remains considerably wider than that of the brick tunnel randomization the unequal allocation procedure with the tightest possible imbalance distribution among all allocation ratio preserving procedures with the same allocation ratio finally the third method the bcr with a preset proportion of maximal forcing mimics the properties of the equal allocation bcr with maximum forcing it approaches the brick tunnel randomization similar to how 1 1 bcr approaches 1 1 pbr with the permuted block size of 2 the equal allocation procedure with the lowest possible imbalance when the bias approaches 1 with minimum forcing the bcr with a preset proportion of maximal forcing approaches complete randomization similar to 1 1 bcr copyright 2017 john wiley sons ltd

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

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

symmetric products of a semistable degeneration of surfaces we explicitly construct a v normal crossing gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toric geometry using this model we calculate the stringy e polynomial of the relative symmetric product we also construct a minimal model of degeneration of hilbert schemes explicitly

stably irrational hypersurfaces of small slopes we show that over any uncountable field of characteristic different from two a very general hypersurface of dimension n 2 and degree at least log 2 n 2 is not stably rational this significantly improves earlier results of kollar and totaro as a byproduct of our proof we obtain new counterexamples to the integral hodge conjecture answering a question of voisin and colliot thelene voisin

affine stanley symmetric functions we define a new family inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of generating functions for w inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i which are affine analogues of stanley symmetric functions we establish basic properties of these functions including symmetry dominance and conjugation we conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine schur functions as applications we show how affine stanley symmetric functions generalize the dual of the k schur functions of lapointe lascoux and morse as well as the cylindric schur functions of postnikov conjecturally affine stanley symmetric functions should be related to the cohomology of the affine flag variety

instanton counting on blowup i 4 dimensional pure gauge theory we give a mathematically rigorous proof of nekrasov s conjecture the integration in the equivariant cohomology over the moduli spaces of instantons on 4 gives a deformation of the seiberg witten prepotential for n 2 susy yang mills theory through a study of moduli spaces on the blowup of 4 we derive a differential equation for the nekrasov s partition function it is a deformation of the equation for the seiberg witten prepotential found by losev et al and further studied by gorsky et al

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities we give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products with parameter t 0 and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras

moduli of mckay quiver representations i the coherent component for a finite abelian group g gl n k we describe the coherent component y of the moduli space m of stable mckay quiver representations this is a not necessarily normal toric variety that admits a projective birational morphism formula obtained by variation of geometric invariant theory quotient as a special case this gives a new construction of nakamura s g hilbert scheme hilbg that avoids the typically highly singular hilbert scheme of g points in formula to conclude we describe the toric fan of y and hence calculate the quiver representation corresponding to any point of y

preserving the allocation ratio at every allocation with biased coin randomization and minimization in studies with unequal allocation the demand for unequal allocation in clinical trials is growing most commonly the unequal allocation is achieved through permuted block randomization however other allocation procedures might be required to better approximate the allocation ratio in small samples reduce the selection bias in open label studies or balance on baseline covariates when these allocation procedures are generalized to unequal allocation special care is to be taken to preserve the allocation ratio at every allocation step this paper offers a way to expand the biased coin randomization to unequal allocation that preserves the allocation ratio at every allocation the suggested expansion works with biased coin randomization that balances only on treatment group totals and with covariate adaptive procedures that use a random biased coin element at every allocation balancing properties of the allocation ratio preserving biased coin randomization and minimization are described through simulations it is demonstrated that these procedures are asymptotically protected against the shift in the rerandomization distribution identified for some examples of minimization with 1 2 allocation the asymptotic shift in the rerandomization distribution of the difference in treatment means for an arbitrary unequal allocation procedure is explicitly derived in the paper

noncommutative enhancements of contractions abstract given a contraction of a variety x to a base y we enhance the locus in y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings d under mild assumptions which hold in the case of 1 crepant partial resolutions admitting a tilting bundle with trivial summand and 2 all contractions with fibre dimension at most one in all cases this produces a global invariant in the crepant setting we then apply this to study derived autoequivalences of x we work generally dropping many of the usual restrictions and so both extend and unify existing approaches in full generality we construct a new endofunctor of the derived category of x by twisting over d and then under appropriate restrictions on singularities give conditions for when it is an autoequivalence we show that these conditions hold automatically when the non isomorphism locus in y has codimension 3 or more which covers determinantal flops and we also control the conditions when the non isomorphism locus has codimension 2 which covers 3 fold divisor to curve contractions

counter example to global torelli problem for irreducible symplectic manifolds a simply connected compact kaehler manifoldx is an irreducible symplectic manifold if there is an everywhere non degenerate holomorphic 2 form on x with h0 x 2x c by definition x has even complex dimension there is a canonical symmetric form qx on h x z which is called the beauvillebogomolov form cf be on the other hand since x is kaehler h x z has a natural hodge structure of weight 2 if two irreducible symplectic manifolds x and y are bimeromorphically equivalent then there is a natural hodge isometry between h x z qx and h y z qy cf o proposition 1 6 2 huy lemma 2 6 debarre de has constructed bimeromorphically equivalent irreducible symplectic manifolds x and y such that x and y are not isomorphic this is a counter example to the following problem

breakthroughs in the theory of macdonald polynomials in 1998 i g macdonald 1 introduced a remarkable new basis for the space of symmetric functions the elements of this basis are denoted where is a partition and p q are two free parameters the s which are now called macdonald polynomials specialize to many of the well known bases for the symmetric functions by suitable choices of the parameters q and t in fact we can obtain in this manner the schur functions the hall littlewood symmetric functions the jack symmetric functions the zonal symmetric functions the zonal spherical functions and the elementary and monomial symmetric functions

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

equivariant autoequivalences for finite group actions the familiar fourier mukai technique can be extended to an equivariant setting where a finite group g acts on a smooth projective variety x in this paper we compare the group of invariant autoequivalences aut db x g with the group of autoequivalences of dg x we apply this method in three cases hilbert schemes on k3 surfaces kummer surfaces and canonical quotients

a sharp lower bound for the log canonical threshold in this note we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function varphi with an isolated singularity at 0 in an open subset of mathbb c n this threshold is defined as the supremum of constants c 0 such that e 2c varphi is integrable on a neighborhood of 0 we relate c varphi to the intermediate multiplicity numbers e j varphi defined as the lelong numbers of dd c varphi j at 0 so that in particular e 0 varphi 1 our main result is that c varphi geqslant sum j 0 n 1 e j varphi e j 1 varphi this inequality is shown to be sharp it simultaneously improves the classical result c varphi geqslant 1 e 1 varphi due to skoda as well as the lower estimate c varphi geqslant n e n varphi 1 n which has received crucial applications to birational geometry in recent years the proof consists in a reduction to the toric case i e singularities arising from monomial ideals

cohomology of the hilbert scheme of points on a surface with values in representations of tautological bundles let x a smooth quasi projective algebraic surface l a line bundle on x let x n the hilbert scheme of n points on x and l n the tautological bundle on x n naturally associated to the line bundle l on x we explicitely compute the image l n of the tautological bundle l n for the bridgeland king reid equivalence d b x n d bn x n in terms of a complex cof sn equivariant sheaves in d bn x n we give moreover a characterization of the image l n l n in terms of of the hyperderived spectral sequence e p q 1 associated to the derived k fold tensor power of the complex c l the study of the sn invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general k fold exterior power of the tautological bundle for the hilbert chow morphism providing danila brion type formulas in these two cases this yields easily the computation of the cohomology of x n with values in l n l n andkl n

the chow groups and the motive of the hilbert scheme of points on a surface abstract we compute the chow motive and the chow groups with rational coefficients of the hilbert scheme of points on a smooth algebraic surface

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

remarks on the derived mckay correspondence for hilbert schemes of points and tautological bundles we study the images of tautological bundles on hilbert schemes of points on surfaces and their wedge powers under the derived mckay correspondence the main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs as an application we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves in particular we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface

a k tableau characterization of k schur functions we study k schur functions characterized by k tableaux proving combinatorial properties such as a k pieri rule and a k conjugation this new approach relies on developing the theory of k tableaux and includes the introduction of a weight permuting involution on these tableaux that generalizes the bender knuth involution this work lays the groundwork needed to prove that the set of k schur littlewood richardson coefficients contains the 3 point gromov witten invariants structure constants for the quantum cohomology ring

birationally rigid hypersurfaces we prove that for n 4 all smooth hypersurfaces of degree n in n are birationally superrigid first discovered in the case n 4 by iskovskikh and manin in a work that started this whole direction of research this property was later conjectured to hold in general by pukhlikov the proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals

a graded representation model for macdonald s polynomials we define doubly graded sn modules rmu for which we conjecture that the multiplicities of irreducible representations in various bi degrees are given by the macdonald coefficients klambdamu assuming one fundamental conjecture the modules rmu can be given several equivalent definitions which we discuss we prove the conjectures in various special cases

symmetric quotient stacks and heisenberg actions for every smooth projective variety x we construct an action of the heisenberg algebra on the direct sum of the grothendieck groups of all the symmetric quotient stacks x n mathfrak s n which contains the fock space as a subrepresentation the action is induced by functors on the level of the derived categories which form a weak categorification of the action

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

the singularities of the principal component of the hilbert scheme of points abstract haiman proved the remarkable result that the isospectral hilbert scheme of points in the plane is normal cohen macaulay and gorenstein this implies the n conjecture and the positivity conjecture for the kostka macdonald coefficients in addition he conjectured that the isospectral hilbert scheme over the principal component of hilb n c d is cohen macaulay for any d n 1 we provide a counterexample to the conjecture

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

on the universal family of hilbert schemes of points on a surface abstract for a smooth quasi projective surface x and an integer n 3 we show that the universal family z n over the hilbert scheme hilb n x of n points has non q gorenstein rational singularities and that the samuel multiplicity at a closed point on z n can be computed in terms of the dimension of the socle we also show that n

vertex operators and hall littlewood symmetric functions abstract we consider vertex operators on space v with a parameter t their components form an associative algebra which is a generalization of the clifford algebra a distinguished orthogonal basis of v is proved to be the hall littlewood symmetric functions we show that kostka foulkes polynomials or certain kazhdan lusztig polynomials for the affine weyl group of type a are matrix coefficients on the space v we also obtain certain generating functions for the product of hall littlewood functions and the kostka foulkes polynomials

representations of rational cherednik algebras with minimal support and torus knots abstract in this paper we obtain several results about representations of rational cherednik algebras and discuss their applications our first result is the cohen macaulayness property as modules over the polynomial ring of cherednik algebra modules with minimal support our second result is an explicit formula for the character of an irreducible minimal support module in type a n 1 for c m n and an expression of its quasispherical part i e the isotypic part of hooks in terms of the homfly polynomial of a torus knot colored by a young diagram we use this formula and the work of calaque enriquez and etingof to give explicit formulas for the characters of the irreducible equivariant d modules on the nilpotent cone for sl m our third result is the construction of the koszul bgg complex for the rational cherednik algebra which generalizes the construction of the koszul bgg resolution from 3 and 21 and the calculation of its homology in type a we also show in type a that the differentials in the koszul bgg complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the cherednik algebra finally our fourth result is the symmetry theorem which identifies the quasispherical components in the representations with minimal support over the rational cherednik algebras h m n s n and h n m s m in fact we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras this symmetry was essentially established in 8 in the spherical case and in 24 in the case gcd m n 1 and it has a natural interpretation in terms of invariants of torus knots

vertex operators and character varieties abstract we prove some combinatorial conjectures extending those proposed in 13 14 the proof uses a vertex operator due to nekrasov okounkov and the first author 4 to obtain a gluing formula for the relevant generating series essentially reducing the computation to the case of cp 1 with three punctures

rational cherednik algebras and hilbert schemes ii representations and sheaves let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring gr u c mathbb c mathfrac h mathfrac h w where is the n th symmetric group using the z algebra construction from gs it is also possible to associate to a filtered h c or u c module hat m on the hilbert scheme hilb n using this technique we study the representation theory of u c and h c and we relate it to hilb n and to the resolution of singularities hilb n mathfrac h mathfrac h w for example we prove the following if c 1 n so that l c triv is the unique one dimensional simple h c module then hat e l c triv mathcal o z n where z n 1 0 is the punctual hilbert scheme if c 1 n k for k in mathbb n then under a canonical filtration on the finite dimensional module l c triv gr e l c triv has a natural bigraded structure that coincides with that on h 0 z n mathscr l k where mathscr l mathcal o hilb n 1 this confirms conjectures of berest etingof and ginzburg beg2 conjectures 7 2 7 3 under mild restrictions on c the characteristic cycle of hat e c equals sum k z where k are kostka numbers and the z are known irreducible components of 1 mathfrak h w

on certain graded sn modules and the q kostka polynomials abstract we derive here a number of properties of the q kostka polynomials k q in particular we obtain a very accessible proof that these polynomials have non negative integer coefficients other monotonicity properties are also derived these results are obtained by studying certain graded s n modules r which afford a character that may be expressed in terms of the k q certain nesting properties of the r which correspond to the dominance order of partitions then translate themselves into combinatorial inequalities involving the k q the modules r have been given an elementary presentation by deconcini and procesi invent math 64 1981 203 219 as rather simple quotients of the polynomial ring q x 1 x 2 x n we show here that their basic properties may also be derived in an entirely elementary manner

brick tunnel randomization for unequal allocation to two or more treatment groups studies with unequal allocation to two or more treatment groups often require a large block size for permuted block allocation this could present a problem in small studies multi center studies or adaptive design dose finding studies in this paper an allocation procedure which generalizes the maximal procedure by berger ivanova and knoll to the case of k 2 treatment groups and any allocation ratio is offered brick tunnel bt randomization requires the allocation path drawn in the k dimensional space to stay close to the allocation ray that corresponds to the targeted allocation ratio specifically it requires the allocation path to be confined to the set of the k dimensional unitary cubes that are pierced by the allocation ray the brick tunnel the important property of the bt randomization is that the transition probabilities at each node within the tunnel are defined in such a way that the unconditional allocation ratio is the same for every allocation step this property is not necessarily met by other allocation procedures that implement unequal allocation

parabolic refined invariants and macdonald polynomials a string theoretic derivation is given for the conjecture of hausel letellier and rodriguez villegas on the cohomology of character varieties with marked points their formula is identified with a refined bps expansion in the stable pair theory of a local root stack generalizing previous work of the first two authors in collaboration with pan haiman s geometric construction for macdonald polynomials is shown to emerge naturally in this context via geometric engineering in particular this yields a new conjectural relation between macdonald polynomials and refined local orbifold curve counting invariants the string theoretic approach also leads to a new spectral cover construction for parabolic higgs bundles in terms of holomorphic symplectic orbifolds

brick tunnel randomization and the momentum of the probability mass the allocation space of an unequal allocation permuted block randomization can be quite wide the development of unequal allocation procedures with a narrower allocation space however is complicated by the need to preserve the unconditional allocation ratio at every step the allocation ratio preserving arp property when the allocation paths are depicted on the k dimensional unitary grid where allocation to the l th treatment is represented by a step along the l th axis l 1 to k the arp property can be expressed in terms of the center of the probability mass after i allocations specifically for an arp allocation procedure that randomizes subjects to k treatment groups in w1 wk ratio w1 wk 1 the coordinates of the center of the mass are w1 i wk i in this paper the momentum with respect to the center of the probability mass expected imbalance in treatment assignments is used to compare arp procedures in how closely they approximate the target allocation ratio it is shown that the two arm and three arm brick tunnel randomizations btr are the arp allocation procedures with the tightest allocation space among all allocation procedures with the same allocation ratio the two arm btr is the minimum momentum two arm arp allocation procedure resident probabilities of two arm and three arm btr are analytically derived from the coordinates of the center of the probability mass the existence of the respective transition probabilities is proven probability of deterministic assignments with btr is found generally acceptable copyright 2015 john wiley sons ltd

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

a positive combinatorial formula for symplectic kostka foulkes polynomials i rows abstract we prove a conjecture of lecouvey which proposes a closed positive combinatorial formula for symplectic kostka foulkes polynomials in the case of rows of arbitrary weight to show this we construct a new algorithm for computing cocyclage in terms of which the conjecture is described our algorithm is free of local constraints which were the main obstacle in lecouvey s original construction in particular we show that our model is governed by the situation in type a this approach works for arbitrary weight and we expect it to lead to a proof of the conjecture in full generality

efficient generation of constrained block allocation sequences selection bias is a potential problem especially in trials where blinding is not possible randomization with a constrained block of large size provides better protection from selection bias than using a sequence of blocks of small size with the same maximum imbalance we propose an algorithm that efficiently generates constrained block allocation sequences and we describe two clinical trials in which such a constrained randomization was used

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

dynamic randomization and a randomization model for clinical trials data randomization models are useful in supporting the validity of linear model analyses applied to data from a clinical trial that employed randomization via permuted blocks here a randomization model for clinical trials data with arbitrary randomization methodology is developed with treatment effect estimators and standard error estimators valid from a randomization perspective a central limit theorem for the treatment effect estimator is also derived as with permuted blocks randomization a typical linear model analysis provides results similar to the randomization model results when roughly unit effects display no pattern over time a key requirement for the randomization inference is that the unconditional probability that any patient receives active treatment is constant across patients when this probability condition is violated the treatment effect estimator is biased from a randomization perspective most randomization methods for balanced 1 to 1 treatment allocation satisfy this condition however many dynamic randomization methods for planned unbalanced treatment allocation like 2 to 1 do not satisfy this constant probability condition and these methods should be avoided copyright 2012 john wiley sons ltd

macdonald polynomials as characters of cherednik algebra modules we prove that macdonald polynomials are characters of irreducible cherednik algebra modules

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

hall littlewood polynomials and vector bundles on the hilbert scheme abstract let e be the bundle defined by applying a polynomial functor to the tautological bundle on the hilbert scheme of n points in the complex plane by a result of haiman 5 the cech cohomology groups h i e vanish for all i 0 it follows that the equivariant euler characteristic with respect to the standard two dimensional torus action has nonnegative integer coefficients in the torus variables z 1 z 2 because they count the dimensions of the weight spaces of h 0 e we derive a formula for this euler characteristic using residue formulas for the euler characteristic coming from the description of the hilbert scheme as a quiver variety 13 14 we evaluate this expression using jing s hall littlewood vertex operator with parameter z 1 7 and a new vertex operator formula given in proposition 1 below we conjecture that the summand in this formula is a polynomial in z 1 with nonnegative integer coefficients a special case of which was known to lascoux and schutzenberger 8

schur function identities their t analogs and k schur irreducibility abstract we obtain general identities for the product of two schur functions in the case where one of the functions is indexed by a rectangular partition and give their t analogs using vertex operators we study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of schur functions and also provides a convenient environment for studying the macdonald polynomials we use our identities to prove that the vertex operators leave such subspaces invariant we finish by showing that these operators act trivially on the k schur functions thus leading to a concept of irreducibility for these functions

some combinatorial identities related to commuting varieties and hilbert schemes in this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the hilbert scheme of points in the plane