projective functors for algebraic groups we investigate the regular subquotient category introduced by soergel in w soergel on the relation between intersection cohomology and representation theory in positive characteristic j pure appl algebra 152 1 3 2000 311 335 a detailed study of projective functors enables one to relate those categories for semi simple algebraic group g and its subgroup schemata grt as an application we derive some information about the characters of tilting modules for g

some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

monotone methods for a finite difference system of reaction diffusion equation with time delay abstract the aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction diffusion equations with time delays the iterative schemes lead to computational algorithms as well as existence uniqueness and upper and lower bounds of the solution an application to a diffusive logistic equation with time delay is given

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

asymptotic periodicity in diffusive logistic equations with discrete delays has been the object of intensive analysis by numerous authors l 6 as a model of single species population growth it is known that time delay has a tendency to produce oscillations 2 5 61 in 1 which is nonoscillatory when r 0 in an autonomous delay system modeling two competing species time delays can even lead to chaotic behavior 4 if the diffusion is taken into account in the growth rate as well then time delay may cause random and blow up effects in the parabolic logistic delay equation 7 nonlinear periodic diffusion equations arise naturally in population models s 12 where the birth and death rates rates of diffusion rates of interaction and environmental carrying capacities are periodic on a seasonal scale in this paper we study the following periodicparabolic logistic equation with instantaneous and delay effects duct x 3t au t x u t x a t x m x u t x

solving the two dimensional constant quantum yang baxter equation a detailed analysis of the constant quantum yang baxter equation rk1k2j1j2 rl1k3k1j3rl2l3k2k3 rk2k3j2j3 rk1l3j1k3rl1l2k1k2 in two dimensions is presented leading to an exhaustive list of its solutions the set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the grobner basis methods each solution was then transformed into a canonical form based on the various trace matrices of r for final elimination of duplicates and subcases if we use homogeneous parametrization the solutions can be combined into 23 distinct cases modulo the well known c p and t reflections and rotations and scalings r q q r q q 1

solving the yang baxter like matrix equation for rank two matrices abstract let a p q t where p and q are two n 2 complex matrices of full column rank such that det q t p 0 and so 0 is a semisimple eigenvalue of a with multiplicity n 2 we solve the quadratic matrix equation a x a x a x completely

categories with projective functors we introduce a notion of a category with full projective functors it encodes certain common properties of categories appearing in representation theory of lie groups lie algebras and quantum groups we describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors

a family of solutions of the yang baxter equation a new method to construct involutive non degenerate set theoretic solutions xn r n of the yang baxter equation for any positive integer n from a given solution x r is presented furthermore the permutation group g xn r n associated with the solution xn r n is isomorphic to a subgroup of g x r and in many cases g xn r n g x r

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

graded decomposition numbers for the blob algebra abstract we give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity

lower central series and free resolutions of hyperplane arrangements if m is the complement of a hyperplane arrangement and a h m k is the cohomology ring of m over a field of characteristic 0 then the ranks k of the lower central series quotients of 1 m can be computed from the betti numbers bii dimtor a k k i of the linear strand in a minimal free resolution of k over a we use the cartan eilenberg change of rings spectral sequence to relate these numbers to the graded betti numbers b dimtor e a k j of a minimal resolution of a over the exterior algebra e from this analysis we recover a formula of falk for 3 and obtain a new formula for 4 the exact sequence of low degree terms in the spectral sequence allows us to answer a question of falk on graphic arrangements and also shows that for these arrangements the algebra a is koszul if and only if the arrangement is supersolvable we also give combinatorial lower bounds on the betti numbers b 1 of the linear strand of the free resolution of a over e if the lower bound is attained for i 2 then it is attained for all i 2 for such arrangements we compute the entire linear strand of the resolution and we prove that all components of the first resonance variety of a are local for graphic arrangements which do not attain the lower bound unless they have no braid sub arrangements we show that b 1 is determined by the number of triangles and k4 subgraphs in the graph

highest weight sl_2 categorifications ii structure theory this paper continues the study of highest weight categorical sl2 actions initi ated in part i we start by refining the definition given there and showing that all examples considered in part i are also highest weight categorifications in the refined sense then we prove that any highest weight sl2 categorification can be filtered in such a way that the suc cessive quotients are so called basic highest weight sl2 categorifications for a basic highest weight categorification we determine minimal projective resolutions of standard objects we use this in particular to examine the structure of tilting objects in basic categorifications and to show that the ringel duality is given by the rickard complex we apply some of these structural results to categories o for cyclotomic rational cherednik algebras we finish by discussing open problems

monotone iterates for solving nonlinear integro parabolic equations of volterra type the paper deals with numerical solving of nonlinear integro parabolic problems based on the method of upper and lower solutions a monotone iterative method is constructed existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given construction of initial upper and lower solutions is discussed numerical experiments are presented

a generalized koszul theory and its relation to the classical theory abstract let a i 0 a i be a graded locally finite k algebra where a 0 is a finite dimensional algebra whose finitistic dimension is 0 in this paper we develop a generalized koszul theory preserving many classical results and show an explicit correspondence between this generalized theory and the classical theory applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described

koszul duality and equivalences of categories let a and a be dual koszul algebras by positselski a filtered algebra u with gr u a is koszul dual to a differential graded algebra a d we relate the module categories of this dual pair by a hom adjunction this descends to give an equivalence of suitable quotient categories and generalizes work of beilinson ginzburg and soergel

numerical methods for nonlinear integro parabolic equations of fredholm type abstract this paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction diffusion convection equations under either linear or nonlinear boundary conditions the discrete approximation of the problem is based on the finite difference method and the computation of the finite difference solution is by the method of upper and lower solutions three types of quasi monotone reaction functions are considered and for each type a monotone iterative scheme is obtained each of these iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system this monotone convergence leads to an existence uniqueness theorem as well as a computational algorithm for the computation of the solution an error estimate between the computed approximations and the true finite difference solution is obtained for each iterative scheme these error estimates are given in terms of the strength of the reaction function and the effect of diffusion convection and are independent of the true solution applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

solving the yang baxter like matrix equation for a class of elementary matrices let u and v be two n dimensional vectors in the complex field such that v t u 0 and let a i u v t we find all the solutions of the quadratic matrix equation a x a x a x in various situations with the help of a spectral perturbation result for rank one updated matrices two numerical examples are given to demonstrate the effectiveness of the method including an application of the result to a 4 4 equation related to the classic yang baxter equation

complete commuting solutions of the yang baxter like matrix equation for diagonalizable matrices let a be a square matrix that is diagonalizable we find all the commuting solutions of the quadratic matrix equation a x a x a x by taking advantage of the jordan form structure of a together with the help of a well known theorem on the uniqueness of a solution to sylvester s equation two special classes of the given matrix a are further investigated including circular matrices and those that are equal to some of their powers moreover all the non commuting solutions are constructed when a is a householder matrix based on a spectral perturbation result

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

on commuting solutions of the yang baxter like matrix equation abstract let a be an arbitrary square matrix and its jordan canonical form is p 1 a p j diag j 1 1 j q q with p an invertible matrix 1 q are different eigenvalues of matrix a the commuting solution problem of the matrix equation a x a x a x is equivalent to the problem j i i y i y i j i i j i i 2 y i j i i y i 2 with y p 1 x p diag y 1 y q we give the structures of the commuting solutions y i in special toeplitz forms based on them we construct new matrices h 1 2 related to the commuting solutions then we propose a method of solving all the commuting yang baxter like solutions by which all solutions can be obtained step by step by recursively solving matrix equations in two cases i 0 or i 0 with respect to the i th jordan block j i i

retractability of set theoretic solutions of the yang baxter equation abstract it is shown that square free set theoretic involutive non degenerate solutions of the yang baxter equation whose associated permutation group referred to as an involutive yang baxter group is abelian are retractable in the sense of etingof schedler and soloviev this solves a problem of gateva ivanova in the case of abelian iyb groups it also implies that the corresponding finitely presented abelian by finite groups called the structure groups are poly z groups secondly an example of a solution with an abelian involutive yang baxter group which is not a generalized twisted union is constructed this answers in the negative another problem of gateva ivanova the constructed solution is of multipermutation level 3 retractability of solutions is also proved in the case where the natural generators of the iyb group are cyclic permutations moreover it is shown that such solutions are generalized twisted unions

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

primary ideals associated to the linear strands of lascoux s resolution and syzygies of the corresponding irreducible representations of the lie superalgebra gl m n abstract we investigate equivariant koszul duality between primary ideals i a b of s s m m n associated to rectangular young diagrams a b and the corresponding atypical irreducible mixed supertensor representations x a b of gl m n in characteristic zero we show i a b to be h 0 s x a b of the koszul dual s x a b and we compute all the higher cohomology of s x a b to be direct sums of i a r b r with multiplicities being given by coefficients of gauss polynomials utilizing this we are able to describe the equivariant syzygies of x a b over s m m n and determine the homological dimension of i a b over s

two block springer fibers of types c and d a diagrammatic approach to springer theory we explain an elementary topological construction of the springer representation on the homology of topological springer fibers of types c and d in the case of nilpotent endomorphisms with two jordan blocks the weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types b and d we determine the decomposition of the representations into irreducibles and relate our construction to classical springer theory as an application we obtain presentations of the cohomology rings of all two block springer fibers of types c and d moreover we deduce explicit isomorphisms between the kazhdan lusztig cell modules attached to the induced trivial module and the irreducible specht modules in types c and d

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

asymptotic stability for an integrodifferential reaction diffusion system abstract a reaction diffusion system involving an integrodifferential linear parabolic equation and a nonlinear ordinary differential equation is studied under third type boundary conditions such a system may be used to describe the evolution of a class of man environment epidemics conditions are given for the existence and the asymptotic stability of a unique nontrivial equilibrium solution by using monotone iteration techniques

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

spectral solutions of the yang baxter matrix equation abstract let a be an arbitrary square matrix we find a collection of solutions of the matrix equation of a x a x a x in terms of the spectral projectors

multiplicities of second cell tilting modules abstract we consider three related representation theories that of a quantum group at a complex root of unity that of an almost simple algebraic group over an algebraically closed field of prime characteristic and that of the symmetric group the main results of this paper concern multiplicities in modular tilting modules we prove a formula valid for type a n 2 d n e 6 e 7 e 8 and g 2 giving the multiplicities of indecomposable tilting modules with highest weight in an explicitly described set of alcoves the proof relies on quantizations of the modular tilting modules and is an application of a recent result by soergel describing the quantum tilting modules in terms of hecke algebra combinatorics in fact the set of alcoves just mentioned corresponds to the second largest kazhdan lusztig cell in the affine weyl group associated to our root system giving rise to the phrase second cell tilting modules the groethendieck group comes with two bases that of weyl modules and that of tilting modules based on the multiplicity formula we give the coefficients of second cell tilting modules in any weyl module of independent interest is an application of the modular multiplicity formula we determine the dimensions of a set of simple representations of the symmetric group over a field of characteristic p the dimension formula covers simple modules parametrized by partitions n 1 n n with either n 1 n n 1 p n 2 or n 2 n n p n 2 this generalizes a result of mathieu lett math phys 38 1 1996 23 32 as well as a recent result by jensen and mathieu on three lines representations of the symmetric group in preparation further it proves in part a conjecture by mathieu in analysis on homogeneous spaces and representation theory of lie groups okayama kyoto 1997 math soc japan tokyo 2000 pp 145 212

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

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

explicit solutions of the yang baxter like matrix equation for a diagonalizable matrix with spectrum contained in 1 α 0 abstract let a cl l be a diagonalizable matrix with spectrum contained in the set 1 0 in this paper we derive a general and explicit expression for the solutions x of the yang baxter like matrix equation a x a x a x the idea involves partitioning the matrices to discuss four block matrix equations and utilizing the eigen properties of the matrices obtained when a is an idempotent matrix the result generates the formula obtained in mansour et al 2017 we give examples to illustrate the validity of the results obtained in this note

numerical solving nonlinear integro parabolic equations by the monotone weighted average method the paper deals with numerical solving nonlinear integro parabolic problems of volterra type based on the weighted average method a monotone iterative method is presented construction of initial upper and lower solutions is given existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given numerical experiments are presented

iterative methods for finding commuting solutions of the yang baxter like matrix equation abstract the main goal of this paper is the numerical computation of solutions of the so called yang baxter like matrix equation a x a x a x where a is a given complex square matrix two novel matrix iterations are proposed both having second order convergence a sign modification in one of the iterations gives rise to a third matrix iteration strategies for finding starting approximations are discussed as well as a technique for estimating the relative error one of the methods involves a very small cost per iteration and is shown to be stable numerical experiments are carried out to illustrate the effectiveness of the new methods

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

monotone iterative methods for numerical solutions of nonlinear integro elliptic boundary problems abstract the aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction diffusion convection equations under linear boundary conditions the boundary value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations the construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function and the iterative schemes are presented for each of the three types of quasimonotone functions an application of the monotone iterations including some numerical results is given to a modified logistic diffusive equation where the kernel in the integral term may be positive negative or changing sign in its domain

commuting solutions of the yang baxter like matrix equation for a class of rank two updated matrices abstract let a i p q t where p and q are two n 2 complex matrices of full column rank and det q t p 0 all the commuting solutions with a of the quadratic matrix equation a x a x a x have been founded