thompson s group f is not almost convex abstract we show that thompson s group f does not satisfy cannon s almost convexity condition ac n for any positive integer n with respect to the standard generating set with two elements to accomplish this we construct a family of pairs of elements at distance n from the identity and distance 2 from each other which are not connected by a path lying inside the n ball of length less than k for increasingly large k our techniques rely upon fordham s method for calculating the length of a word in f and upon an analysis of the generators geometric actions on the tree pair diagrams representing elements of f

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

refinements of the littlewood richardson rule in the prequel to this paper we showed how results of mason involving a new combinatorial formula for polynomials that are now known as demazure atoms characters of quotients of demazure modules called standard bases by lascoux and schutzenberger could be used to dene a new basis for the ring of quasisymmetric functions we call quasisymmetric schur functions qs functions for short in this paper we develop the combinatorics of these polynomials futher by showing that the product of a schur function and a demazure atom has a positive expansion in terms of demazure atoms we use these techniques together with the fact that both a qs function and a demazure character have explicit expressions as a positive sum of atoms to obtain the expansion of a product of a schur function with a qs function demazure character as a positive sum of qs functions demazure characters our formula for the coecients in the expansion of a product of a demazure character and a schur function into demazure characters is similar to known results and includes in particular the famous littlewood richardson rule for the expansion of a product of schur functions in terms of the schur basis a composition weak composition with n parts is a sequence of n positive nonnegative integers respectively a partition is a composition whose parts are monotone nonincreasing if is a weak composition composition or partition we let denote the number of parts of throughout this article is a weak composition with n while and denote compositions and partitions respectively with n n the polynomials in this paper schur functions demazure atoms and characters qs functions depend on a

a filtration on the cohomology rings of regular nilpotent hessenberg varieties let n be a positive integer the main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent hessenberg variety in gl n mathbb c b such that its associated graded ring has graded pieces i e homogeneous components isomorphic to rings which are related to the cohomology rings of hessenberg varieties in gl n 1 mathbb c b showing the inductive nature of these rings in previous work the first two authors together with abe and masuda gave an explicit presentation of these cohomology rings in terms of generators and relations we introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy this allows us to derive our filtration in addition we obtain the following three corollaries first we give an inductive formula for the poincare polynomial of these varieties second we give an explicit monomial basis for the cohomology rings of regular nilpotent hessenberg varieties with respect to the presentation mentioned above third we derive a basis of the set of linear relations satisfied by the images of the schubert classes in the cohomology rings of regular nilpotent hessenberg varieties finally our methods and results suggest many directions for future work in particular we propose a definition of hessenberg schubert polynomials in the context of regular nilpotent hessenberg varieties which generalize the classical schubert polynomials we also outline several open questions pertaining to them

isoperimetric inequalities and the homology of groups an isoperimetric function for a finitely presented group g bounds the number of defining relations needed to prove that a word w g 1 in terms of the length of w suppose g f r is a finitely presented group where f is a finitely generated free group and r is the normal closure of a finite set of relators for each w r we define the area w to be the least number of uses of the given relators needed to prove that w g 1 the isoperimetric function g n is defined to be the maximum of the areas of all the words in r of length n one compares the size of such functions using a suitable order 1 on functions different presentations give rise to equivalent isoperimetric functions where is the equivalence relation determined by 1 interest in such functions was stimulated by gromov 13 who characterized word hyperbolic groups as those with linear isoperimetric function that is g n n further information was obtained by cannon et al 8 who showed that automatic groups satisfy g 1 n and that asynchronously automatic groups satisfy g 1 2 they also showed the free nilpotent group of class 2 and rank 2 has g n n gersten 10 used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups he exploited the fact that for a torsion free one relator groupg the usual 2 dimensional complex having fundamental group isomorphic to g is aspherical the general technique for giving upper bounds for g n is to describe a solution to the word problem for g and analyze its use of defining relations it seems quite difficult to establish lower bounds for g n in this manner in this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for g instead of trying to analyze r directly we consider some of its quotient groups for which additional machinery is available for instance we consider for each w r its image w r f in r r f and define a suitable centralized isoperimetric function g n such that g n 1 g n using the hopf formula it can be shown that r r f is a direct sum of the second homology group h2 g z with a free abelian

measuring the tameness of almost convex groups a 1 combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points x in the 1 skeleton of the cayley 2 complex associated to the presentation we define two functions radial and ball tameness flnctions that measure how efficiently a 1 combing moves away from the identity these functions are geometric in the sense that they are quasi isometry invariants we show that a group is almost convex if and only if the radial tameness function is bounded by the identity function hence almost convex groups as well as certain generalizations of almost convex groups are contained in the quasi isometry class of groups admitting linear radial tameness functions

autostackability of thompson s group f abstract the word problem for thompson s group f has a solution but it remains unknown whether f is automatic or has a finite or regular convergent terminating and confluent rewriting system we show that the group f admits a natural extension of these two properties namely autostackability and we give an explicit bounded regular convergent prefix rewriting system for f

the dehn function of the generalized thompson group is quadratic abstract guba 8 proved that the dehn function of the thompson group f is quadratic in this paper we apply his idea and extend his result to the generalized thompson group f l for any integer l 3

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

trees parking functions syzygies and deformations of monomial ideals for a graph g we construct two algebras whose dimensions are both equal to the number of spanning trees of g one of these algebras is the quotient of the polynomial ring modulo certain monomial ideal while the other is the quotient of the polynomial ring modulo certain powers of linear forms we describe the set of monomials that forms a linear basis in each of these two algebras the basis elements correspond to g parking functions that naturally came up in the abelian sandpile model these ideals are instances of the general class of monotone monomial ideals and their deformations we show that the hilbert series of a monotone monomial ideal is always bounded by the hilbert series of its deformation then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals in some cases these resolutions coincide with scarf resolutions we prove several formulas for hilbert series of monotone monomial ideals and investigate when they are equal to hilbert series of deformations in the appendix we discuss the abelian sandpile model

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

lattice diagram polynomials and extended pieri rules abstract the lattice cell in the i 1 st row and j 1 st column of the positive quadrant of the plane is denoted i j if is a partition of n 1 we denote by ij the diagram obtained by removing the cell i j from the french ferrers diagram of we set ij det x p j i y q j i n i j 1 where p 1 q 1 p n q n are the cells of ij and let m ij be the linear span of the partial derivatives of ij the bihomogeneity of ij and its alternating nature under the diagonal action of s n gives m ij the structure of a bigraded s n module we conjecture that m ij is always a direct sum of k left regular representations of s n where k is the number of cells that are weakly north and east of i j in we also make a number of conjectures describing the precise nature of the bivariate frobenius characteristic of m ij in terms of the theory of macdonald polynomials on the validity of these conjectures we derive a number of surprising identities in particular we obtain a representation theoretical interpretation of the coefficients appearing in some macdonald pieri rules

covers of algebraic varieties iii the discriminant of a cover of degree 4 and the trigonal construction in c e a general structure theorem for gorenstein covers of degrree d has been proved such a result has been used for characterizing covers of degree 3 4 and 5 see c e and cs respectively of algebraic varieties more precisely in section 4 resp 3 of c e it is proved that the total space x of each gorenstein cover e x y of degree 4 esp 3 is obta ned inside a suit bl p ojective bundle 7r 11 y of rank 2 resp 1 as the b se locus of a pencil relativ conics resp

tame combing and almost convexity conditions we give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation but which are not minimally almost convex on a standard generating set namely we explicitly construct such combings for thompson s group f and the baumslag solitar groups bs 1 p with p 3 in order to make this construction for thompson s group f we significantly expand the understanding of the cayley complex of this group with respect to the standard finite presentation in particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2 cells adjacent to edges that do not lie on normal form paths

algorithms and topology of cayley graphs for groups autostackability for finitely generated groups is defined via a topological property of the associated cayley graph which can be encoded in a finite state automaton autostackable groups have solvable word problem and an effective inductive procedure for constructing van kampen diagrams with respect to a canonical finite presentation a comparison with automatic groups is given another characterization of autostackability is given in terms of prefix rewriting systems every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix closed set of normal forms is also autostackable as a consequence the fundamental group of every closed 3 manifold with any of the eight possible uniform geometries is autostackable

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

k theory formulas for orthogonal and symplectic orbit closures abstract the complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits we study two families of polynomials introduced by wyser and yong representing the k theory classes of the closures of these orbits our polynomials are analogous to the grothendieck polynomials representing k classes of schubert varieties and we show that like grothendieck polynomials they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property we show that the same polynomials represent the equivariant k classes of symmetric and skew symmetric analogues of knutson and miller s matrix schubert varieties we derive explicit expressions for these polynomials in special cases including a pfaffian formula relying on a more general degeneracy locus formula of anderson finally we show that taking an appropriate limit of our representatives recovers the k theoretic schur q functions of ikeda and naruse

combinatorial properties of thompson s group f we study some combinatorial consequences of blake fordham s theorems on the word metric of thompson s group f in the standard two generator presentation we explore connections between the tree pair diagram representing an element w of f its normal form in the infinite presentation its word length and minimal length representatives of it we estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates in addition we explore some properties of the cayley graph of f with respect to the two generator finite presentation namely we exhibit the form of dead end elements in this cayley graph and show that it has no deep pockets finally we discuss a simple method for constructing minimal length representatives for strictly positive or negative words

the dehn function of richard thompson s group f is quadratic we prove that the dehn function that is the smallest isoperimetric function of the r thompson s group f is quadratic

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

on positivity of line bundles on enriques surfaces we study linear systems on enriques surfaces we prove rationality of seshadri constants of ample line bundles on enriques surfaces and provide lower bounds on these numbers we show the nonexistence of k very ample line bundles on enriques surfaces of degree 4k 4 for k 1 thus answering an old question of ballico and sommese introduction in this note we answer a question posed by ballico and sommese 1 see also terakawa 14 concerning the existence of k very ample line bundles of degree 4k 4 on enriques surfaces we show that such line bundles exist if and only if k 0 and that in fact the degree of a k very ample line bundle on an enriques surface is subject to a much more restrictive bound theorem 2 4 this has strong consequences for the local positivity of line bundles on enriques surfaces we address this questions in section 3 in particular we show that seshadri constants of ample line bundles on enriques surfaces are always rational 1 auxiliary material and notation the basic reference for this paper is the monograph 6 by cossec and dolgachev here we briefly recall properties of enriques surfaces needed for our considerations first of all an enriques surface is a surface y of kodaira dimension zero with no differential forms in particular ho ky 0 and the canonical bundle ky is a 2torsion element in pico y i e 2ky oy to alleviate notation it is convenient to use the following convention in the sequel we make frequent use of vanishing theorems and statements on adjoint line bundles since the canonical divisor ky is numerically trivial the adjoint line bundle ky l has the same numerical properties as l in particular if l is nef and big ample then so is ky l by the nakai moishezon criterion so in order to get vanishing statements for l we apply vanishing theorems to ky l and use the fact that 2ky is trivial in the sequel we do so without comment and hope to cause no confusion this way received by the editors june 1 2000 2000 mathematics subject classification primary 14j28 secondary 14c20 14e25

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

orbit closures in the enhanced nilpotent cone we study the orbits of g gl v in the enhanced nilpotent cone where is the variety of nilpotent endomorphisms of v these orbits are parametrized by bipartitions of n dimv and we prove that the closure ordering corresponds to a natural partial order on bipartitions moreover we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of kostka polynomials defined by shoji finally we make a connection with kato s exotic nilpotent cone in type c proving that the closure ordering is the same and conjecturing that the intersection cohomology is the same but with degrees doubled

on the quotient ring by diagonal invariants for a finite coxeter group w and its reflection representation we find the character and hilbert series for a quotient ring of by an ideal containing the w invariant polynomials without constant term this confirms conjectures of haiman

bases for coordinate rings of conjugacy classes of nilpotent matrices abstract a general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix this conjecture is proven when the partition given by the transpose jordan type of the nilpotent matrix is a hook or has two parts

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

hall littlewood vertex operators and generalized kostka polynomials abstract a family of vertex operators that generalizes those given by jing for the hall littlewood symmetric functions is presented these operators produce symmetric functions related to the poincare polynomials referred to as generalized kostka polynomials in the same way that jing s operator produces symmetric functions related to kostka foulkes polynomials these operators are then used to derive commutation relations and new relations involving the generalized kostka coefficients such relations may be interpreted as identities in the gl n c equivariant k theory of the nullcone

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

hilbert functions of sn stable artinian gorenstein algebras abstract we describe the graded characters and hilbert functions of certain graded artinian gorenstein quotients of the polynomial ring which are also representations of the symmetric group specifically we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form

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

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

on the picard group of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a non singular projective surface s with h s d s h2 s cs 0 and 2ks 0 the unramitied double cover defined by ks is a k3 surface r a non singular projective surface with hi r d r 0 kr 0 illusie has shown i that the group of divisors modulo numerical equivalence is isomorphic to the enriques lattice u e 8 1 where u and e s 1 denote respectively the unique even unimodular lattices of index of inertia 1 1 and 0 8 the purpose of this note is to use this isomorphism to study the picard group of s we prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for s and r for example we prove the following results

decomposition of green polynomials of type a and springer modules for hooks and rectangles abstract this paper deals with graded representations of the symmetric group on the cohomology ring of flags fixed by a unipotent matrix we consider a combinatorial property called the coincidence of dimension of the graded representations and give an interpretation in terms of representation theory of the symmetric group in the case where the corresponding partition of the unipotent matrix is a hook or a rectangle the interpretation is equivalent to a recursive formula of green polynomials at roots of unity

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

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

congruence properties of q analogs abstract using group actions and generating functions we derive various arithmetic properties of the q binomial coefficients and q stirling numbers these include recurrence relations and computation of residues modulo a cyclotomic polynomial we also obtain periodicity and non periodicity results

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

cyclage catabolism and the affine hecke algebra abstract we identify a subalgebra h n of the extended affine hecke algebra h n of type a the subalgebra h n is a u analogue of the monoid algebra of s n z 0 n and inherits a canonical basis from that of h n we show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n which we term positive affine tableaux pat we then exhibit a cellular subquotient r 1 n of h n that is a u analogue of the ring of coinvariants c y 1 y n e 1 e n with left cells labeled by pat that are essentially standard young tableaux with cocharge labels multiplying canonical basis elements by a certain element h n corresponds to rotations of words and on cells corresponds to cocyclage we further show that r 1 n has cellular quotients r that are u analogues of the garsia procesi modules r with left cells labeled by a pat version of the catabolizable tableaux we give a conjectural description of a cellular filtration of h n the subquotients of which are isomorphic to dual versions of r under the perfect pairing on r 1 n this turns out to be closely related to the combinatorics of the cells of h n worked out by shi lusztig and xi and we state explicit conjectures along these lines we also conjecture that the k atoms of lapointe lascoux and morse 2003 9 and the r catabolizable tableaux of shimozono and weyman 2000 20 have cellular counterparts in h n we extend the idea of atom copies from lapointe lascoux and morse 2003 9 to positive affine tableaux and give descriptions mostly conjectural of some of these copies in terms of catabolizability

reye constructions for nodal enriques surfaces a classical reye congruence x is an enriques surface of rational equivalence class 3 7 in the grassmannian g 1 3 of lines of p 3 x is the locus of lines of p 3 which are included in two quadrics of w web of quadrics a generaiization to g 1 t is given 1 for each t 2 there exist enriques surfaces x of class t 3t 2 in g 1 t 2 the determinant of the dual of the universal bundle on x is o x 2e r k x with e isolated elliptic curve r 2 2 e r t 3 x parameterizes lines of p t which are included in a codimension 2 subsystem of w w linear system of quadrics of dimension 2 t the paper includes a description of the variety of trisecant lines to a smooth enriques surface of degree 10 p 5

a mckay correspondence for reflection groups we construct a noncommutative desingularization of the discriminant of a finite reflection group g as a quotient of the skew group ring a s g if g is generated by order 2 reflections then this quotient identifies with the endomorphism ring of the reflection arrangement a g viewed as a module over the coordinate ring sg of the discriminant of g this yields in particular a correspondence between the nontrivial irreducible representations of g to certain maximal cohen macaulay modules over the coordinate ring sg these maximal cohen macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement a g viewed as a module over sg we identify some of the corresponding matrix factorizations namely the so called logarithmic co residues of the discriminant

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

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

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

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

descent representations and multivariate statistics combinatorial identities on weyl groups of types a and b are derived from special bases of the corresponding coinvariant algebras using the garsia stanton descent basis of the coinvariant algebra of type a we give a new construction of the solomon descent representations an extension of the descent basis to type b using new multivariate statistics on the group yields a refinement of the descent representations these constructions are then applied to refine well known decomposition rules of the coinvariant algebra and to generalize various identities