the partial schur multiplier of a group abstract refining the technique worked out in previous papers we give a characterization up to equivalence of the factor sets of partial projective representation of a group g over an algebraically closed field k in terms of equalities satisfied by this allows one to conclude that any component of the partial schur multiplier p m g is an epimorphic image of a direct power of k moreover it is shown that any component of p m g is an epimorphic image of the component p m g g g of the equivalence classes of the totally defined partial factor sets examples with cyclic g are considered in particular the total component p m g g g is determined when g is an arbitrary finite cyclic group in this case p m g g g is a direct power of k

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

infinite iteration of matrix semigroups ii structure theorem for arbitrary semigroups up to aperiodic morphism where 0 is defined by a a 8 a a here intuitively a is easily globally computed because concatenation of two strings is a transparent multiplication and a a i s surely special however here 0 need not be line the unsolvability of the word problem for semigroups is one way to state this we shall formulate it differently later in fact using the 25 0021 8693 86 3 00

expansions free inverse semigroups and schutzenberger product in this paper we shall present a new construction of the free inverse monoid on a set x contrary to the previous constructions of 9 11 our construction is symmetric and originates from classical ideas of language theory the ingredients of this construction are the free group on x and the relation that associates to a word w of the free monoid on x the set of all pairs u v such that uv w it follows at once from our construction that the free inverse monoid on x can be naturally embedded into the schutzenberger product of two free groups of basis x we shall also give some connections with the theory of expansions as developed by rhodes and birget 2 3

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

minimum entropy quantizers and permutation codes amplitude quantization and permutation encoding are two approaches to efficient digitization of analog data it has been proven that they are equivalent in the sense that their optimum rate versus distortion performances are identical reviews of the aforementioned results and of work performed in the interim by several investigators are presented equations which must be satisfied by the thresholds of the minimum entropy quantizer that achieves a prescribed mean r th power distortion are derived and an iterative procedure for solving them is developed it is shown that these equations often have many families of solutions in the case of the laplacian distribution for which we had previously shown that quantizers with uniformly spaced thresholds satisfy the equations when r 2 other families of solutions with nonuniform spacing are exhibited what had appeared to be a discrepancy between the performances of optimum permutation codes and minimum entropy quantizers is resolved by the resulting optimum quantizers which span all entropy rates from zero to infinity

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

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

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

on optimal permutation codes permutation codes are vector quantizers whose codewords are related by permutations and in one variant sign changes asymptotically as the vector dimension grows optimal variant i permutation code design is identical to optimal entropy constrained scalar quantizer ecsq design however contradicting intuition and previously published assertions there are finite block length permutation codes that perform better than the best ones with asymptotically large length thus there are variant i permutation codes whose performances cannot be matched by any ecsq along similar lines a new asymptotic relation between variant i and variant ii permutation codes is established but again demonstrated to not necessarily predict the performances of short codes simple expressions for permutation code performance are found for memoryless uniform and laplacian sources the uniform source yields the aforementioned counterexamples

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

partial cohomology of groups abstract we develop a cohomology theory of groups based on partial actions and explore its relation with the partial schur multiplier as well as with cohomology of inverse semigroups

bifree locally inverse semigroups as expansions abstract in studia sci math hungar 41 2004 39 58 we constructed for a completely simple semigroup c an expansion s c which is isomorphic to the birget rhodes expansion c pr j algebra 120 1989 284 300 if c is a group analogous to the fact proven in j algebra 120 1989 284 300 that c pr contains a copy of the free inverse semigroup in case c is the free group on x we show that s c contains a copy of the bifree locally inverse semigroup if c is the bifree completely simple semigroup on x as a consequence among other things we obtain a new proof of a result due to f pastijn trans amer math soc 273 1982 631 655 which says that each locally inverse semigroup divides a perfect rectangular band of e unitary inverse monoids

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

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

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

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

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

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

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

two algebraic approaches to variants of the concatenation product we extend an existing approach of the bideterministic concatenation product of languages aiming at the study of three other variants unambiguous left deterministic and right deterministic such an approach is based on monoid expansions the proofs are purely algebraic and use another approach based on properties on the kernel category of a monoid relational morphism without going through the languages this gives a unified fashion to deal with all these variants and allows us to better understand the connections between these two approaches finally we show that local finiteness of an m variety is transferred to the m varieties corresponding to these variants and apply the general results to the m variety of idempotent and commutative monoids

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

group theory via global semigroup theory abstract groups are shown to be special homomorphic images of inverse semigroups that are residually finite actually every element has only finitely many elements j above this also leads to a new approach to the burnside problem these results extend an earlier paper j c birget and j rhodes j pure appl algebra 32 1984 249 287 but can be read independently our goal here is not so much to prove theorems about inverse semigroups as to demonstrate the usefulness of the constructions of j c birget and j rhodes j pure appl algebra 32 1984 249 287

permutation codes for the laplacian source corresp permutation codes for the laplacian source are developed the performance of these codes is evaluated and compared with other quantizers and the rate distortion function it is shown that there is a bit rate region in which the permutation codes outperform certain single sample quantizers

blocks of restricted rational cherednik algebras for g m d n abstract we study the dunkl opdam subalgebra of the rational cherednik algebra for wreath products at t 0 and use this to describe the block decomposition of restricted rational cherednik algebras for g m d n

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

reflection functors for quiver varieties and weyl group actions we introduce reflectionfunctors on quiver varieties they are hyper kahler isometries between quiver varieties with different parameters related by elements in the weyl group the definition is motivated by the origial reflection functor given by bernstein gelfand ponomarev 1 but they behave much nicely they are isomorphisms and satisfy the weyl group relations as an application we define weyl group representations of homology groups of quiver varieties they are analogues of slodowy s construction of springer representations of the weyl group

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

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

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

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

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

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

infinite iteration of matrix semigroups i structure theorem for torsion semigroups given these conditions we must precisely define egc special and line we begin with egc if x is a finite nonempty set x the free monoid generated by x is by definition egc because intuitively concatenation of two finite strings is a transparent multiplication however subsemigroups and especially homomorphisms of egcs need not be egcs furthermore if s is egc and some simple operations applied to s gives s it is reasonable to say s is also egc thus by definition if s is egc and r b x a s then s r s the rees matrix semigroup over s see definition 4 16 is also egc note that the local structure e g green relations egg box pictures etc of r s can be easily computed from the local structure of s and from r for similar reasons if 2 and s are egc r b x a s z a and b z are actions and r bz a r b za for all b e b z e z a e a then by definition z r s see definition 4 16 is also egc 422 0021 8693186 3 00

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