on the algebraic area of lattice walks and the hofstadter model we consider the generating function of the algebraic area of lattice walks evaluated at a root of unity and its relation to the hofstadter model in particular we obtain an expression for the generating function of the nth moments of the hofstadter hamiltonian in terms of a complete elliptic integral evaluated at a rational function this in turn gives us both exact and asymptotic formulas for these moments

area distribution of two dimensional random walks on a square lattice the algebraic area probability distribution of closed planar random walks of length n on a square lattice is considered the generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded a particular case generalizes the q binomial theorem to the case of three addends the distribution fits the levy probability distribution for brownian curves with its first order 1 n correction quite well even for n rather small

a diagrammatic approach to hecke algebras of the reflection equation hecke algebra deformations of the wreath product cd sn arise in solutions to the yang baxter equations with boundary the reflection equation we use a simple diagrammatic approach to construct bases for these algebras we hence introduce quotient algebras suitable for constructing physical solutions to the reflection equation with a well defined thermodynamic large n limit for all d and n we determine the generic representation theory of these algebras developing a formalism suitable for the analysis of corresponding q spin chain hamiltonian and transfer matrix spectra

algebraic areas enclosed by brownian curves on bounded domains using constrained path integrals and perturbation theory the authors study the algebraic area a enclosed by closed brownian curves on various domains such as rectangles or strips in the limit of infinitely long curves the probability distribution p a is shown to be gaussian the standard deviation a2 1 2 is simply expressed in terms of the length scales of the problem

on decomposition matrices for graded algebras abstract we study the relations between the decomposition matrix of a module over a graded algebra and the decomposition matrix of its restriction to the grading subalgebra we show that under appropriate hypotheses if one of the decomposition matrices is stable unitriangular then so is the other a particular case is when one of them is square unitriangular then both are using these results we describe the decomposition matrices of the hecke algebras of the complex reflection groups g de e n with respect to the ones of the ariki koike algebras or the decomposition matrices of q schur algebras of type dn with respect to the ones of q schur algebras of type bn

isomorphisms between simple modules of degenerate cyclotomic hecke algebras abstract we give explicit isomorphisms between simple modules of degenerate cyclotomic hecke algebras defined via various cellular bases a special case gives a generalized mullineux involution in the degenerate case

blocks of cyclotomic hecke algebras abstract this paper classifies the blocks of the cyclotomic hecke algebras of type g r 1 n over an arbitrary field rather than working with the hecke algebras directly we work instead with the cyclotomic schur algebras the advantage of these algebras is that the cyclotomic jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic schur algebras we obtain an explicit description of the blocks by analyzing the combinatorics of jantzen equivalence we remark that a proof of the classification of the blocks of the cyclotomic hecke algebras was announced in 1999 unfortunately cox has discovered that this previous proof is incomplete

hecke algebras of finite type are cellular let mathcal h be the one parameter hecke algebra associated to a finite weyl group w defined over a ground ring in which bad primes for w are invertible using deep properties of the kazhdan lusztig basis of mathcal h and lusztig s a function we show that mathcal h has a natural cellular structure in the sense of graham and lehrer thus we obtain a general theory of specht modules for hecke algebras of finite type previously a general cellular structure was only known to exist in types an and bn

slim cyclotomic q schur algebras abstract we construct a new basis for a slim cyclotomic q schur algebra s m n r via symmetric polynomials in jucys murphy operators of the cyclotomic hecke algebra h m r we show that this basis labelled by matrices is not the double coset basis when h m r is the hecke algebra of a coxeter group but coincides with the double coset basis for the corresponding group algebra the hecke algebra at q 1 as further applications we then discuss the cyclotomic schur weyl duality at the integral level this also includes a category equivalence and a classification of simple objects

an analogue of row removal for diagrammatic cherednik algebras we prove an analogue of james donkin row removal theorems for diagrammatic cherednik algebras this is one of the first results concerning the graded decomposition numbers of these algebras over fields of arbitrary characteristic as a special case our results yield a new reduction theorem for graded decomposition numbers and extension groups for cyclotomic q schur algebras

schur weyl reciprocity between quantum groups and hecke algebras of type g p p n abstract let u q gl m p be the quantized universal enveloping algebra of gl m p let be the automorphism of u q gl m p which is defined on generators by e i e i m f i f i m k j k j m for any i z p m z 0 m p 1 m and any j z p m z let h p p n be the hecke algebra of type g p p n with parameters q e where e is a primitive pth root of unity in this paper we establish a schur weyl reciprocity between h p p n and a twisted tensor product of u q gl m p and the group algebra for a cyclic group of order p by using the results in j hu j algebra 274 2004 446 490

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

oscillation of the solutions of parabolic differential equations of neutral type sufficient conditions are obtained for the oscillation of the solutions of linear parabolic differential equations of neutral type of the form t u si 1m l i t u x t t i a t u si 1sa i ft u x t p i p x t u si 1kp i x t u x t s i x t e wx 0 where w is a bounded domain in r n with a piecewise smooth boundary

on the action of ariki koike algebras on tensor space the action of the ariki koike algebra on tensor space is used to show an isomorphism theorem between the ariki koike algebra and a direct sum of matrix rings over tensorproducts of smaller ariki koike algebras as an application we get schur weyl duality for the action of the ariki koike algebra and the levi subalgebra of the q schur algebra when separation conditions holds mit hilfe der operation der ariki koike algebra auf dem tensorraum wird ein isomorphismus zwischen der ariki koike algebra und einer direkten summe von matrizenringen uber tensorprodukten kleinerer ariki koike algebren gezeigt als anwendung ergibt sich das die ariki koike algebra und die leviunteralgebra der q schur algebra schur weyl dualitat erfullen falls die separationsbedingung erfullt ist

representations of ariki koike algebras and grobner shirshov bases in this paper we investigate the structure of ariki koike algebras and their specht modules using grobner shirshov basis theory and combinatorics of young tableaux for a multipartition lambda we find a presentation of the specht module s lambda given by generators and relations and determine its grobner shirshov pair as a consequence we obtain a linear basis of s lambda consisting of standard monomials with respect to the grobner shirshov pair we show that this monomial basis can be canonically identified with the set of cozy tableaux of shape lambda

modular representations of hecke algebras of type g p p n hecke algebras for the complex reflection groups g r p n where p 1 and p r were introduced in the work of ariki j algebra 177 1995 164 185 broue and malle asterisque 212 1993 119 189 in this paper we consider modular representation theory for these algebras in the case where r p we assume that the field k contains a primitive pth root of unity e our method is to study the restrictions of specht modules s for hecke algebras hp n of type g p 1 n with parameters q 1 e ep 1 suppose that f q e 0 in k see 4 7 for definition of f q e for any multipartition 1 p of n we prove that s hp p n s k hp p n for any 1 k p 1 where k k 1 k 2 p 1 2 k and if k is the smallest positive integer such that k hence k p we explicitly decompose s hp p n into a direct sum of p k smaller hp p n submodules with the same dimensions as a result when f q e 0 in k we show that hp p n is split over k and get a complete classification of all the absolutely irreducible hp p n modules this generalizes earlier work of c pallikaros j algebra 169 1994 20 48 and j hu manuscripta math 108 2002 409 430 on hecke algebras of type dn which are included as a special case of our main results

the schur algebra we study a class of endomomorphism algebras of certain q permutation modules over the hecke algebra of type b whose summands involve both parabolic and quasi parabolic subgroups and prove that these algebras are integrally free and quasi hereditary and are stable under base change some consequences for decomposition numbers are discussed the notion of a q schur algebra was introduced by dipper and james dj2 who used these algebras to parametrize the irreducible representations of the finite general linear groups in non describing characteristics with hindsight these algebras had already appeared earlier in an entirely different quantum group context ji inspired by physics in ji jimbo considered the endomorphism algebras of tensor spaces as hecke algebra modules in his context a q schur algebra can be viewed as a quotient of the quantized enveloping algebra associated to gln in pw these algebras were shown to be quasi hereditary the quasi heredity property is an embodiment in classical algebra of the geometric derived category stratification exhibited by perverse sheaves ps it means more applications can be deduced from a ring theoretic point of view see e g dps3 and the possibility is raised of even deeper results in the future as suggested by cps2 certainly these algebras play a central role in the representation theories of the finite and quantum general linear groups naturally one asks are there such algebras for types other than a our paper ds showed that there were similar quasi hereditary quotients of quantized enveloping algebras for all types of root systems however no connection with hecke algebras and finite groups of lie type was found there this paper aims at the same question and constructs possible algebras directly from hecke algebras hence from finite groups of lie type we restrict attention to the type b case imitating the definition of a q schur algebra we introduce the notion of a q schur algebra these algebras are the endomorphism algebras of certain modules over the hecke algebra of type b called tensor spaces whose summands involve not only parabolic subgroups but also quasi parabolic subgroups of the weyl group of type b a main result of our paper shows that qschur algebras are quasi hereditary we speculate that similar constructions exist for other classical types only type d remains actually since type c is equivalent received by the editors march 3 1997 and in revised form october 28 1998 2000 mathematics subject classification primary 20c08 20g05 20c33 the authors would like to thank arc for support under the large grant a69530243 as well as nsf and the universities of virginia and new south wales for their cooperation the first author also thanks the newton institute at cambridge for its hospitality c 2000 american mathematical society

the number of simple modules for the hecke algebras of type g r p n with an appendix by xiaoyi cui abstract we derive a parameterization of simple modules for the cyclotomic hecke algebras of type g r p n with p 1 and n 3 over fields of any characteristic coprime to p we give explicit formulas for the number of simple modules over these cyclotomic hecke algebras

on the decomposition numbers of the hecke algebra of type dn when n is even abstract let n 4 be an even integer let k be a field with char k 2 and q an invertible element in k such that i 1 n 1 1 q i 0 in this paper we study the decomposition numbers over k of the iwahori hecke algebra h q d n of type d n we obtain some equalities which relate its decomposition numbers with certain schur elements and the decomposition numbers of various iwahori hecke algebras of type a with the same parameter q when char k 0 this completely determine all of its decomposition numbers the main tools we used are the morita equivalence theorem established in j hu a morita equivalence theorem for hecke algebra h q d n when n is even manuscripta math 108 2002 409 430 and certain twining character formulae of weyl modules over a tensor product of two q schur algebras

algebraic area enclosed by random walks on a lattice we compute the moments of the area enclosed by an n steps random walk on a 2d lattice we consider separately the cases where the walk comes back to the origin or not we also compute for both cases the characteristic function at order

proof of the modular branching rule for cyclotomic hecke algebras abstract we prove the modular branching rule of the cyclotomic hecke algebras which has remained open

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

cyclotomic solomon algebras this paper introduces an analogue of the solomon descent algebra for the complex reflection groups of type g r 1 n as with the solomon descent algebra our algebra has a basis given by sums of distinguished coset representatives for certain reflection subgroups we explicitly describe the structure constants with respect to this basis and show that they are polynomials in r this allows us to define a deformation or q analogue of these algebras which depends on a parameter q we determine the irreducible representations of all of these algebras and give a basis for their radicals finally we show that the direct sum of cyclotomic solomon algebras is canonically isomorphic to a concatenation hopf algebra

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

a necessary and sufficient condition for oscillation of neutral type hyperbolic equations abstract in the present paper a necessary and sufficient condition is obtained for oscillation of the solutions of neutral type linear hyperbolic differential equations of the form u tt x t l a i u tt x t i u x t j b j t u x j t cu x t k c k u x t k 0 x t 0 g where is a bounded domain in r n with a piecewise smooth boundary and u x t ni 1uxtxt x t

a result on the bruhat order of a coxeter group fix a coxeter group w s the terminology an expression of iv means an element of w being written as a product of elements of s we say an expression w of w is reduced if w is a product of l w elements of s given two expressions s s2 sk and tl t2 t of w with si tje s we write s1s sk tl t t if k m and s t for all i and write s1s2 sk f t t t if otherwise we say that two expressions g and g are congruent if g e g thus we have g e g g g we say g is a subexpression of an expression g if g xg y for some expressions x y of w 510 0021 8693 90 x 3 0

on the semisimplicity of the cyclotomic brauer algebras ii abstract in this paper we give a necessary and sufficient condition for the semisimplicity of cyclotomic brauer algebras b m n of types g m 1 n with m 2 this generalizes h rui a criterion on the semisimple brauer algebras j combin theory ser a 111 2005 78 88 1 2 1 3 and h rui m si a criterion on the semisimple brauer algebras ii j combin theory ser a 113 2006 1199 1203 2 5 on brauer algebras

schur weyl duality for infinitesimal q schur algebras sq 2 r 1 abstract using the result of s r doty d k nakano k m peters polynomial representations of frobenius kernels of gl 2 in contemp math vol 194 1996 pp 57 67 s konig c xi when is a cellular algebra quasi hereditary math ann 315 1999 281 293 we prove that a non semisimple infinitesimal schur algebra s 2 r 1 is not cellular furthermore we determine the structure of the endomorphism ring of tensor space as a module for the infinitesimal schur algebra s 2 r 1 up to morita equivalence both results generalize to the quantum case

statistics of two dimensional random walks the cyclic sieving phenomenon and the hofstadter model we focus on the algebraic area probability distribution of planar random walks on a square lattice with and l2 steps right left up and down we aim in particular at the algebraic area generating function evaluated at a root of unity when both and multiples of q in the simple case of staircase walks a geometrical interpretation of in terms of the cyclic sieving phenomenon is illustrated then an expression for which is relevant to the stembridge case is proposed finally the related problem of evaluating the nth moments of the hofstadter hamiltonian in the commensurate case is addressed

the jantzen sum formula for cyclotomic schur algebras the cyclotomic q schur algebra was introduced by dipper james and mathas in order to provide a new tool for studying the ariki koike algebra we here prove an analogue of jantzen s sum formula for the cyclotomic q schur algebra among the applications is a criterion for certain specht modules of the ariki koike algebras to be irreducible

non vanishing gram determinants for cyclotomic nazarov wenzl and birman murakami wenzl algebras abstract in this paper we use the method in rui and si 2011 35 to give a necessary and sufficient condition on non vanishing gram determinants for cyclotomic nw and cyclotomic bmw algebras over an arbitrary field equivalently we give a necessary and sufficient condition for each cell module of such algebras being equal to its simple head over an arbitrary field

area distribution for directed random walks we study the probability distribution for the area under a directed random walk in the plane the walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed this model captures some of the qualitative features of earthquakes avalanches and other self organized critical phenomena in one dimension by finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4 3

distribution of the area enclosed by a two dimensional random walk in a disordered medium the asymptotic probability distribution for a brownian particle wandering in a two dimensional plane with random traps to enclose the algebraic area a by time t is calculated using the instanton technique

cellular structures on hecke algebras of type b abstract the aim of this paper is to gather and try to unify several approaches for the modular representation theory of hecke algebras of type b we attempt to explain the connections between geck s cellular structures coming from kazhdan lusztig theory with unequal parameters and ariki s theorem on the canonical basis of the fock spaces

borelic pairs for stratified algebras abstract we determine all values of the parameters for which the cell modules form a standard system for a class of cellular diagram algebras including partition brauer walled brauer temperley lieb and jones algebras for this we develop and apply a general theory of finite dimensional algebras with borelic pairs the theory is also applied to give new uniform proofs of the cellular and quasi hereditary properties of the diagram algebras and to construct quasi hereditary 1 covers in the sense of rouquier with exact borel subalgebras in the sense of konig another application of the theory leads to a proof that auslander dlab ringel algebras admit exact borel subalgebras

a necessary and sufficient condition for oscillation of parabolic equations with several delays a necessary and sufficient condition for oscillation of parabolic equations with several delays is obtained

a topological problem in polymer physics configurational and mechanical properties of a random walk enclosing a constant are the authors show how a random walk in a plane constrained to enclose a given area can be used to approximately represent the properties of an entangled polymer molecule the statistical mechanical properties of the loop are calculated exactly and the distribution function for the enclosed areas is found for the case of a random walk with free ends joined by a straight line segment the distribution function is given by the cauchy distribution this implies that the area has statistical fractal properties but does not have a mean for a genuinely closed random walk a mean exists but the distribution of areas is not fractal the spatial and mechanical properties of the constrained configurations have also been calculated analytically if the unrestricted coil can be regarded as an entropic spring of zero natural length then the area constrained configurations behave qualitatively like springs with a finite natural length the deformation behaviour also shows both softening and hardening dependent on the area imposed

exact random walk distributions using noncommutative geometry using the results obtained by the non commutative geometry techniques applied to the harper equation we derive the areas distribution of random walks of length n on a two dimensional square lattice for large n taking into account finite size contributions

an algorithm for the computation of the decomposition matrices for ariki koike algebras abstract we give a purely combinatorial algorithm for the computation of the decomposition matrices for ariki koike algebras when the parameters are powers of the same root of unity

h oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments in this paper sufficient conditions are established for h oscillation of solutions of a class of vector hyperbolic differential equations with deviating arguments where h is a unit vector in r m the main results are illustrated by two examples

tilting modules for cyclotomic schur algebras this paper investigates the tilting modules of the cycloto mic q schur alge bras the young modules of the ariki koike algebras and the interconnections between them the main tools used to understand the tilting modules are contragredient duality and the specht filtrations and dual specht filtrations of cert ain permutation modules sur prisingly weyl filtrations which are in general more power ful than specht filtrations play only a secondary role

on the number of constituents of induced modules of ariki koike algebras abstract we examine the crystal graph of the sl e module arising from an sl e categorification to study the defining endo functors of the categorification this yields lower bounds on the number of irreducible constituents of certain objects we use ariki s categorification result on ariki koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules in particular this will imply reducibility of every induced module

stratifying endomorphism algebras associated to hecke algebras abstract let g be a finite group of lie type and let k be a field of characteristic distinct from the defining characteristic of g in studying the non describing representation theory of g the endomorphism algebra s g k end kg j ind g p j k plays an increasingly important role in type a by work of dipper and james s g k identifies with a q schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups this paper presents the first systematic study of the structure and homological algebra of these algebras for g of arbitrary type because s g k has a reinterpretation as a hecke endomorphism algebra it may be analyzed using the theory of hecke algebras its structure turns out to involve new applications of kazhdan lusztig cell theory in the course of this work we prove two stratification conjectures about coxeter group representations made by e cline b parshall and l scott mem amer math soc 591 1996 and we formulate a new conjecture about the structure of s g k we verify this conjecture here in all rank 2 examples

on tensor spaces over hecke algebras of type bn abstract let w b n be the weyl group of type b n and h b n be the associated iwahori hecke algebra in this paper we study the n tensor space v n where dim v 2 m with natural actions introduced in r m green hyperoctahedral schur algebras j algebra 192 1997 418 438 of w b n and of h b n for each composition 1 m of n let e be the corresponding initial basis element of v n see 3 8 for definition we show that if d is a distinguished right coset representative of s in w b n then the action of the natural basis element t d on e coincides with the permutation action of d up to a scalar as an application we prove that the n tensor space decomposes at the integral level into a direct sum of some permutation modules over hecke algebra h b n with respect to certain standard parabolic subalgebras

on cellular algebras with jucys murphy elements abstract we study analogues of jucys murphy elements in cellular algebras arising from repeated jones basic constructions examples include brauer and bmw algebras and their cyclotomic analogues

on the classification of finite dimensional irreducible modules for affine bmw algebras in this paper we classify the finite dimensional irreducible modules for affine bmw algebra over an algebraically closed field with arbitrary characteristic

the mirabolic hecke algebra abstract the iwahori hecke algebra of the symmetric group is the convolution algebra of gl n invariant functions on the variety of pairs of complete flags over a finite field considering convolution on the space of triples of two flags and a vector we obtain the mirabolic hecke algebra r n which had originally been described by solomon in this paper we give a new presentation for r n which shows that it is a quotient of a cyclotomic hecke algebra as defined by ariki and koike from this we recover the results of siegel about the representations of r n we use jucys murphy elements to describe the center of r n and to give a gl structure on the grothendieck group of the category of its representations giving mirabolic analogues of classical results about the iwahori hecke algebra we also outline a strategy towards a proof of the conjecture that the mirabolic hecke algebra is a cellular algebra

morita equivalence for cyclotomic bmw algebras abstract in this paper we determine explicitly the singular parameters for cyclotomic bmw algebras b r n over an arbitrary field equivalently we give a criterion for b r n being morita equivalent to the direct sum of some ariki koike algebras this generalizes konig and xi s work on brauer algebras 14 theorem 7 3 and our previous work on brauer and birman murakami wenzl algebras 20 proposition 2 15

quasi parabolic subgroups of g m 1 r abstract we study the properties of cosets and double cosets of quasi parabolic subgroups via tableaux we characterize these double cosets by n n arrays of non negative integer sequences for each double coset we construct and count all shortest and longest representatives and find their lengths we also find the number of double cosets in g m 1 r of each pair of quasi parabolic subgroups

a conjecture for q decomposition matrices of cyclotomic v schur algebras abstract the jantzen sum formula for cyclotomic v schur algebras yields an identity for some q analogues of the decomposition matrices of these algebras we prove a similar identity for matrices of canonical bases of higher level fock spaces we conjecture then that those matrices are actually identical for a suitable choice of parameters in particular we conjecture that decomposition matrices of cyclotomic v schur algebras are obtained by specializing at q 1 some transition matrices between the standard basis and the canonical basis of a fock space

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

a two dimensional non markovian random walk leading to anomalous diffusion abstract exact solutions are rare for non markovian random walk models even in 1d and much more so in 2d here we propose a 2d genuinely non markovian random walk model with a very rich phase diagram such that the motion in each dimension can belong to one of 3 categories i subdiffusive ii superdiffusive or iii normally diffusive the main advance reported here is a different method and the consequent physical insight for analytically solving this model simpler non markovian models such as levy walks have been solved in 2d but it is not clear if the method of solution could be made to work for more complicated models such as the one studied here we also report the exact solutions for the first two moments of the random walk propagator along with the complete phase diagram the latter is surprisingly rich and admits diverse diffusion regimes finally we discuss these results in the context of theoretical underpinnings as well as possible applications

the degenerate analogue of ariki s categorification theorem we explain how to deduce the degenerate analogue of ariki s categorification theorem over the ground field mathbb c as an application of schur weyl duality for higher levels and the kazhdan lusztig conjecture in finite type a we also discuss some supplementary topics including young modules tensoring with sign tilting modules and ringel duality

necessary and sufficient conditions for oscillation of robin boundary problems of neutral functional parabolic equations some necessary and sufficient conditions for oscillation of the solutions of robin boundary problems of neutral parabolic equations with several deviating arguments are obtained

canonical basic sets in type bn more than 10 years ago dipper james and murphy developed the theory of specht modules for hecke algebras of type bn more recently using lusztig s a function geck and rouquier showed how to obtain parametrizations of the irreducible representations of hecke algebras of any finite type in terms of so called canonical basic sets for certain values of the parameters in type bn combinatorial descriptions of these basic sets were found by jacon based on work of ariki and foda leclerc okado thibon welsh here we consider the canonical basic sets for all the remaining choices of the parameters

distribution of the area enclosed by a plane random walk the authors derive an exact expression for the distribution function for the algebraic area enclosed by a plane closed random walk using a continuous model from the start their result has a different analytical form but is numerically close to an expression for the same quantity of a discrete model derived by brereton and butler 1987

matrix units and generic degrees for the ariki koike algebras this paper uses the murphy basis of the ariki koike algebras to explicitly construct a complete set of primitive idempotents when these algebras are semisimple and q 1 as a consequence we obtain an explicit wedderburn basis for the ariki koike algebras finally we use these idempotents to compute the generic degrees of the ariki koike algebras

algebraic areas enclosed by 2d brownian curves in random media we study closed brownian trajectories when the brownian particle is subjected to a random potential for a poissonian repulsive potential the enclosed algebraic area a is a gaussian variable that scales like t3 4 when t the length of the curve goes to infinity this is intermediate between the situations where the particle is allowed to wander i everywhere on the plane a t and ii only on a bounded domain a t1 2 for the lloyd model we show that the probability distribution p a is the same as in the absence of disorder this surprising result is related to some peculiarities of the cauchy law

categorifications and cyclotomic rational double affine hecke algebras varagnolo and vasserot conjectured an equivalence between the category mathcal o for crdaha s and a subcategory of an affine parabolic category o of type a we prove this conjecture as applications we prove a conjecture of rouquier on the dimension of simple modules of crdaha s and a conjecture of chuang miyachi on the koszul duality for the category mathcal o of crdaha s