the 2 braid group and garside normal form we investigate the relation between the garside normal form for positive braids and the 2 braid group defined by rouquier inspired by work of brav and thomas we show that the garside normal form is encoded in the action of the 2 braid group on a certain categorified left cell module this allows us to deduce the faithfulness of the 2 braid group in finite type we also give a new proof of paris theorem that the canonical map from the generalized braid monoid to its braid group is injective in arbitrary type

alternative proof for the localization of sinai s walk we give an alternative proof of the localization of sinai s random walk in random environment under weaker hypothesis than the ones used by sinai moreover we give estimates that are stronger than the one of sinai on the localization neighborhood and on the probability for the random walk to stay inside this neighborhood

coxeter elements and kazhdan lusztig cells abstract by the correspondence between coxeter elements of a coxeter system w s and the acyclic orientations of the coxeter graph we study some properties of elements in the set c0 w we show that when w is of finite affine or hyperbolic type any w c0 w satisfies w lr w j with l wj j m w for some j s now assume that w is of finite or affine type we give an explicit description for all the distinguished involutions d of w with d l w for some w e w which verifies a conjecture proposed in jian yi shi adv sci china math 3 1990 79 98 conjecture 8 10 in our case we show that any left cell of w containing some element of c0 w is left connected which verifies a conjecture of lusztig ryoshi hotta ed in problems from the conference on algebraic groups and representations held at katata august 29 september 3 1983 in our case

farey level separation in mode locking structure of circle mappings abstract we consider various characteristic scales both on the x axis and in the parameter space for critical and near critical circle mappings juxtaposition of different scales for the set of all rational mode lockings exposes the effect of separation into layers this layered structure is shown to reproduce farey tree levels of the corresponding rotation numbers the separation is studied numerically for maps with various orders of singularity and interpreted in terms of coefficients of rotation function expansion

random walks in asymmetric random environments we consider random walks on zd with transition ratesp x y given by a random matrix ifp is a small random perturbation of the simple random walk we show that the walk remains diffusive for almost all environmentsp ifd 2 the result also holds for a continuous time markov process with a random drift the corresponding path space measures converge weakly in the scaling limit to the wiener process for almost everyp

the cells of the affine weyl group c n in a certain quasi split case abstract the affine weyl group c n s can be realized as the fixed point set of the affine weyl group a 2 n 1 s under a certain group automorphism with s s let l be the length function of a 2 n 1 we study the cells of the weighted coxeter group c n l the main results of the paper are to give an explicit description for all the cells of c n l corresponding to the partitions k 1 2 n k and h 21 2 n h 2 for all 1 k 2 n and 2 h 2 n 2 and also for all the cells of c 3 l

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

the joint relations and the set d1 in certain crystallographic groups abstract for any coxeter group w d kazhdan and g lusztig 1979 invent math 53 165 184 introduced some joint relations among the elements of w by which certain representations of w and its associated hecke algebra h are constructed recently in a letter to lusztig t a springer obtained some interesting results concerning the connection between some non zero structural coefficients of h and the joint relations in w when w is a certain crystallographic group he investigated some properties of a subset d 1 of w springer denoted d 1 by f in his letter which plays an important role for the joint relations in w in the present paper we shall develop some results of springer we shall deduce further joint relations in w from some non zero structural coefficients of h in particular we shall make some detailed discussion for the set d 1 this includes the approach to the proof of a conjecture of lusztig concerning the characterization of the set d 1 the connection between the finiteness of d 1 and some joint relations in w the explicit description for some elements of d 1 and the connectedness of the set d 0 d 2 1 see 3 2 and 5 2 for the notations

nilpotent orbits and commutative elements abstract let w be a simply laced coxeter group with generating set s and let w c denote the subset consisting of those elements whose reduced expressions have no substrings of the form sts for any non commuting s t s we give a root system characterization of w c and in the case where w corresponds to a finite weyl group show that w c is a union of spaltenstein springer steinberg cells the latter is valid also for affine weyl groups of type a but not for type d or e

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

identification of simple representations for affine q schur algebras abstract we define an explicit action of the quantum loop algebra u q gl n on the tensor space and show that these actions are compatible with the natural embedding u q gl n u q gl n 1 as an application we identify simple representations of affine q schur algebras s n r c arising from simple polynomial representations of u q gl n frenkel and mukhin 2002 11 with those arising from simple modules of the affine hecke algebra h r c of type a rogawski 1985 26

left connectedness of some left cells in certain coxeter groups of simply laced type abstract let w be an irreducible finite or affine weyl group of simply laced type we show that any w w with a w 6 satisfies condition c w x w j y for some x y w and some j s with w j finite and l w j a w see 0 1 0 2 for the notation w j w j l w and a w we also show that if l is a left cell of w all of whose elements satisfy condition c then the distinguished involution d l of w in l satisfies d l z 1 z z 1 w j z for any z w j z e min l with j l z see 1 6 for the notation z 1 z and 0 3 for l z e min l and e l verifying a conjecture of mine in j y shi a survey on the cell theory of affine weyl groups adv sci china math 3 1990 79 98 conjecture 8 10 in our case if e l e min l then we show that the left cell l is left connected verifying a conjecture of lusztig in our case

left cells in the affine weyl group of typec4 we find a representative set of left cells of the affine weyl groupwaof typec4as well as its left cell graphs by applying an algorithm this algorithm was designed in my previous paper tohoku j math 46 1994 105 124 it is reformulated and improved in a more efficient form here these representatives of left cells are presented as the vertices of so called essential graphs so that the generalized invariants of left cells ofwaare actually described explicitly the latter almost characterize the left cells ofwa some comments and conjectures are proposed on cells of affine weyl groups mostly for the case of typecl l 2

an invariance principle for isotropic diffusions in random environment we investigate in this work the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of brownian motion when the space dimension is three or more we prove an invariance principle as well as transience our methods also apply to questions of homogenization in random media

a counter example to a conjecture of lusztig abstract in lusztig 2003 7 lusztig defined two functions a and a on a coxeter group w and conjectured that they are the same in the present paper we give a counter example to lusztig s conjecture then we propose a new conjecture for the description of two sided cells of w which can be regarded as an improvement of lusztig s conjecture some examples are given to support our conjecture

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

central limit theorem and recurrence for random walks in bistochastic random environments we prove the annealed central limit theorem for random walks in bistochastic random environments on zd with zero local drift the proof is based on a dynamicist s interpretation of the system and requires a much weaker condition than the customary uniform ellipticity moreover recurrence is derived for d 2

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

complete devil s staircase fractal dimension and universality of mode locking structure in the circle map it is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil s staircase at the onset of chaos the complementary set to the stability intervals is a cantor set of fractal dimension d 0 87 this exponent is found to be universal for a large class of functions

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

the laurent polynomials my ws in the hecke algebra with unequal parameters abstract let w s be a coxeter system and h the associated hecke algebra with unequal parameters the laurent polynomials m y w s and p y w for y w w and s s play an important role in the representations of h we study the properties of m y w s and p y w the relations among them as well as with the left right and two sided cells of w

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

lusztigʼs a function for coxeter groups of rank 3 abstract we show that lusztig s a function of a coxeter group is bounded if the rank of the coxeter group is 3

symmetric random walks in random environments we consider a random walk on thed dimensional lattice d where the transition probabilitiesp x y are symmetric p x y p y x different from zero only ify x belongs to a finite symmetric set including the origin and are random we prove the convergence of the finite dimensional probability distributions of normalized random paths to the finite dimensional probability distributions of a wiener process and find our an explicit expression for the diffusion matrix

some cells in the weighted coxeter group c n l 2n 1 abstract the fixed point set of the affine weyl group a 2 n 1 s under a certain group automorphism with s s can be considered as the affine weyl group c n s then we study the cells of the weighted coxeter group c n l 2 n 1 with l 2 n 1 the length function of a 2 n 1 we give an explicit description for all the cells of c n l 2 n 1 corresponding to the partitions k 1 2 n 2 k and h 2 n 2 h for any 1 k 2 n 2 and n 1 h 2 n and also for all the cells of c 3 l 7

lagrangian construction for representations of hecke algebras 1 let k be a non archimedean local field with the residue field consisting of q elements according to langlands philosophy irreducible complex representations of a reductive k group g should roughly speaking correspond to homomorphisms of the weil deligne group into the socalled l group let zc g be the iwahori subgroup the category of admissible gkmodules generated by z fixed vectors is known to be equivalent to the category of finite dimensional representations of the hecke algebra h z g z langlands philosophy suggests via that equivalence the following classification of irreducible hecke algebra modules stated in purely complex terms and not involving any local held

laplace operator and random walk on one dimensional nonhomogeneous lattice a classical result of probability theory states that under suitable space and time renormalization a random walk converges to brownian motion we prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice under suitable conditions on the nonhomogeneous medium we prove convergence to brownian motion and explicitly compute the diffusion coefficient the proofs are based on the study of the spectrum of random matrices of increasing dimension

rigorous renormalisation group and disordered systems abstract we consider random walks in a non symmetric random environment we report a recent result based on a renormalisation group approach showing that for d 62 2 these walks are diffusive when the disorder is weak

on two presentations of the affine weyl groups of classical types abstract the main result of the paper is to get the transition formulae between the alcove form and the permutation form of w wa where wa is an affine weyl group of classical type on the other hand we get a new characterization for the alcove form of an affine weyl group element which has a much simpler form compared with that in j y shi j london math soc 35 1987 42 55 as applications we give an affirmative answer to a conjecture of h eriksson and k eriksson preprint 1998 concerning the characterization of the inverse table of w wa we also describe the number s w in terms of the permutation form of w wa

persistent random walks in a one dimensional random environment central limit theorems are obtained for persistent random walks in a onedimensional random environment they also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values

universal scaling in circle maps abstract we discuss scaling renormalization and universality for critical and subcritical circle maps using the structure of the rationals provided by the farey tree we construct a pair of renormalization group operators for circle maps with all values of the winding number in this general formulation the renormalization equations define a universal invariant set which determines all universal quantities for all cubic maps with all irrational winding numbers as expected the manifold of pure rotations defines the subcritical attracting set we show that the invariant set governing the critical circle maps cannot be embedded in less than 3 dimensions however we are able to obtain a differentiable parametrization of the unstable manifold of the critical invariant set which we believe is universal up to lipschitz homeomorphisms the derivative of this function describes the scaling structure of all small gaps in the devil s staircase for all critical maps our work can be viewed as providing numerical evidence for lanford s conjectures on the strange set underlying renormalizations of critical circle maps

the scaling of arnol d tongues for differentiable homeomorphisms of the circle we show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms that the widths of the arnol d tongues in a one parameter family scale asq 3 whenq is the denominator of the rotation number

formulae relating the bernstein and iwahori matsumoto presentations of an affine hecke algebra abstract we consider the anti dominant variants of the elements occurring in the bernstein presentation of an affine hecke algebra h we find explicit formulae for in terms of the iwahori matsumoto generators tw w ranging over the extended affine weyl group of the root system r in the case i r is arbitrary and is a minuscule coweight or ii r is attached to gln and mek where ek is a standard basis vector and m 1 in the above cases certain minimal expressions for play a crucial role such minimal expressions exist in fact for any coweight for gln we give a sheaf theoretic interpretation of the existence of a minimal expression for the corresponding perverse sheaf on the affine schubert variety x t is the push forward of an explicit perverse sheaf on the demazure resolution m x t x t this approach yields for a minuscule coweight of any r or for an arbitrary coweight of gln a conceptual albeit less explicit expression for the coefficient w of the basis element tw in terms of the cohomology of a fiber of the demazure resolution

pro p iwahori hecke algebra and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring h 1 of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring h 1 is finitely generated as a module over its centre these results are proved in 12 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of h 1 to k which is as null as possible will be called null the simple h k modules with a null central character are called supersingular when g gl n we show that each simple h k module of dimension n containing a character of the affine subring h aff is supersingular the proof uses the minimal expressions of haines generalized to h 1 and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 11 and when n 3 rachel ollivier

on birth and death processes in symmetric random environment we prove a limit theorem for a process in a random one dimensional medium which has been considered before as a model for hopping conduction in a disordered medium to the edge between the two integersj and j 1 a rate j 0 is attached these j j integral are taken as independent identically distributed random variables and represent the medium for given values j x t is a markov chain in continuous time which jumps fromj to j 1 and from j 1 toj at the same rate j we show that in many cases there exists normalizing constants y t which tend to oo witht such that the distribution of x t t or more generally of the whole processx st t s 0 converges to a limit as t the limit process is continuous and self similar

a proof of lusztig s conjectures for affine type g2 with arbitrary parameters we prove lusztig s conjectures bf p1 bf p15 for the affine weyl group of type tilde g 2 for all choices of parameters our approach to compute lusztig s mathbf a function is based on the notion of a balanced system of cell representations for the hecke algebra we show that for arbitrary coxeter type the existence of balanced system of cell representations is sufficient to compute the mathbf a function and we explicitly construct such a system in type tilde g 2 for arbitrary parameters we then investigate the connection between kazhdan lusztig cells and the plancherel theorem in type tilde g 2 allowing us to prove bf p1 and determine the set of duflo involutions from there the proof of the remaining conjectures follows very naturally essentially from the combinatorics of weyl characters of types g 2 and a 1 along with some explicit computations for the finite cells

a modified blm approach to quantum affine mathfrak gl _n we introduce a spanning set of beilinson lusztig macpherson type a j r a j for affine quantum schur algebras boldsymbol mathcal s vartriangle n r and construct a linearly independent set a j a j for an associated algebra boldsymbol widehat mathcal k vartriangle n we then establish explicitly some multiplication formulas of simple generators e vartriangle h h 1 mathbf 0 by an arbitrary element a j in boldsymbol widehat mathcal k vartriangle n via the corresponding formulas in boldsymbol mathcal s vartriangle n r and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the ringel hall algebras boldsymbol mathfrak h vartriangle n associated with cyclic quivers this allows us to use the triangular relation between monomial and pbw type bases for boldsymbol mathfrak h vartriangle n established in deng and du adv math 191 276 304 2005 to derive similar triangular relations for boldsymbol mathcal s vartriangle n r and boldsymbol widehat mathcal k vartriangle n using these relations we then show that the subspace boldsymbol mathfrak a vartriangle n of boldsymbol widehat mathcal k vartriangle n spanned by a j a j contains the quantum enveloping algebra mathbf u vartriangle n of affine type a as a subalgebra as an application we prove that when this construction is applied to quantum schur algebras boldsymbol mathcal s n r the resulting subspace boldsymbol mathfrak a vartriangle n is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of mathfrak gl n we conjecture that boldsymbol mathfrak a vartriangle n is a subalgebra of boldsymbol widehat mathcal k vartriangle n

the second lowest two sided cell in an affine weyl group abstract let w a be an irreducible affine weyl group with w 0 the associated weyl group the present paper is to study the second lowest two sided cell qr of w a let n qr be the number of left cells of w a in qr we conjecture that the equality n qr 1 2 w 0 should always hold when w a is either a n 1 n 2 or of rank 4 this equality can be verified by the existing data see 0 3 then the main result of the paper is to prove the inequality n qr 1 2 w 0 in all cases

embeddings of root systems ii permutation root systems abstract this paper studies a class of embeddings of permutation root systems with properties abstracting those of examples of embeddings of linear root systems over commutative rings as considered in the first paper in this series the root system of one coxeter group w regarded as a permutation representation is realized with its roots as sets of roots of another coxeter group w with action induced by an embedding of w in a completion of w

w graph ideals and duality abstract this paper is the continuation of the work in 14 in that paper we generalized the definition of w graph ideal in the weighted coxeter groups and showed how to construct a w graph from a given w graph ideal in the case of unequal parameters in this paper we study the duality and the full w graph for a given w graph ideal we show that there are two modules associated with a given w graph ideal they are connected by a duality map the full w graph includes all the w graph data determined by the dual and contragredient representations our construction closely parallels that of kazhdan and lusztig 6 10 11 which can be regarded as the special case j it also generalizes the work of couillens 2 deodhar 3 4 and douglass 5 corresponding to the parabolic case

isomorphisms between affine hecke algebras of type a2 abstract we prove that two affine hecke algebras h p and h q of type a 2 with nonzero parameters p q c are not isomorphic whenever p q and p q 1 theorem 3 1 however they are isomorphic locally more precisely we prove that the quotient algebras h s q and h t p are isomorphic provided that n s q n t p for type a 2 theorem 2 4

on the basic states of one dimensional disordered structures the purpose of this paper is to study a limit probability distribution of the set of the first eigenvalues 1 2 with a fixed and of the boundary problem on the interval 0 wherea t q t are the random stationary processes particularly the question of the repulsion between the first eigenvalues small energetic levels is studied it has been proved that in the divergent case q t 0 a t 0 levels repulsion exists as for the potential case a t 1 q t 0 there is not any repulsion at all this is one of the main differences between these two cases

presenting affine q schur algebras we obtain a presentation of certain affine q schur algebras in terms of generators and relations the presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine our results extend and rely on the corresponding result for the q schur algebra of the symmetric group which were proved by the first author and giaquinto

conjugacy relation on coxeter elements abstract let w s be an irreducible finitely presented coxeter system the present paper is mainly concerned with a conjugacy relation on coxeter elements in the case where contains just one circle in particular when is itself a circle in the cases where is either a three multiple circle or a circle with three nodes we show that the ss equivalence relation on coxeter elements of w is the same as the w conjugacy relation an explicit formula is given for the characteristic polynomial of a coxeter element in the natural reflection representation of w when is a circle we also give the answers to some questions raised by coleman and extend some results of geck and pfeiffer concerning conjugacy relation in coxeter groups

comparison ofa functions abstract we show agreement of lusztig sa function with thea function previously given by c k fan j amer math soc 10 no 1 1997 139 167

epipelagic representations and invariant theory letg be a reductive p adic group we give a new construction of small depth epipelagic supercuspidal representations ofg k using stable orbits in geometric invariant theory git in contrast to previously known methods this construction works without any restrictions on the residue characteristic p we construct appropriate langlands parameters for these representations with some restrictions on p the git arises from graded moy prasad filtrations which we show are isomorphic to the graded lie algebras whose git was analyzed by vinberg and levy this leads to a classification of stable orbits in moy prasad filtrations as well as epipelagic supercuspidal representations in terms of z regular elliptic automorphisms of the absolute root system of g transferred to the l group of g generates the image of tame inertia under the corresponding wild langlands parameter for unramified groups and sufficiently large p we also classify the moy prasad filtrations which have semi stable orbits which solves the long outstanding problem of classifying non degenerate k types for g k

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

nilpotent ideals in having a fixed class of nilpotence combinatorics and enumeration we study the combinatorics of ad nilpotent ideals of a borel subalgebra of sl n 1 c we provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence we study the relationships between these results and the combinatorics of dyck paths based upon a remarkable bijection between ad nilpotent ideals and dyck paths finally we propose a q t analogue of the catalan number c n these q t catalan numbers count on the one hand ad nilpotent ideals with respect to dimension and class of nilpotence and on the other hand admit interpretations in terms of natural statistics on dyck paths

on effective conductivity on z d lattice we study the effective conductivity e for a random wire problem on the d dimensional cubic lattice d d 2 in the case when random conductivities on bonds are independent identically distributed random variables we give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order in the 2d case using the duality symmetry we also derive the 6th order expansion we compare our results with the bruggeman approximation and show that in the 2d case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms

pattern characterization of rationally smooth affine schubert varieties of type a abstract schubert varieties in finite dimensional flag manifolds g p are a well studied family of projective varieties indexed by elements of the corresponding weyl group w in particular there are many tests for smoothness and rational smoothness of these varieties one key result due to lakshmibai and sandhya is that in type a the smooth schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412 recently there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with g being a kac moody group and w being an affine weyl group or parabolic quotient in this paper we study the case when w is the affine weyl group of type a or the affine permutations we develop the notion of pattern avoidance for affine permutations our main result is a characterization of the rationally smooth schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations

fully commutative elements in the weyl and affine weyl groups abstract let w be a weyl or an affine weyl group and let w c be the set of fully commutative elements in w we associate each w w c to a digraph g w by using g w we give a graph theoretic description for lusztig s a function on w c and describe explicitly all the distinguished involutions of w c the results verify two conjectures in our case one was proposed by myself in adv sci china math 3 1990 79 98 conjecture 8 10 and the other by lusztig in t asai et al open problems in algebraic groups in r hotta ed problems from the conference on algebraic groups and representations katata august 29 september 3 1983

the weighted universal coxeter group and some related conjectures of lusztig abstract lusztig proposed 15 conjectures for any weighted coxeter groups in this study we explicitly describe kazhdan lusztig cells distinguished involutions and the function a for any weighted universal coxeter group we also verify these conjectures for this group

local shimura correspondence it splits over the maximal compact subgroup let t be the corresponding lift of i the iwahori subgroup of g now fix a faithful character of n f let r be the category of smooth representations of which are generated by its t fixed vectors and such that n f acts via also let h be the hecke algebra of i biinvariant compactly supported functions on satisfying f gq f g for z v obj r let v be the space of i fixed vectors then f f l v is a representation of h t according to bernstein and borel 2 this gives the equivalence of r and the category of representations of h in particular this is the case for g whose hecke algebra will be denoted by h g and category by ri g in this paper we describe h in terms of generators and relations as a consequence in the case of simple group for example we find a group g related to g such that h t h g in particular we get the equivalence of r t and

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

embeddings of root systems i root systems over commutative rings abstract this paper gives examples of embeddings of root systems of coxeter groups including sporadic embeddings of standard real root systems in other standard real root systems and for a general coxeter group an embedding of its universal symmetric root system over commutative rings into the standard real root system of a simply laced coxeter group

pro p iwahori hecke ring and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring open image in new window of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring open image in new window is finitely generated as a module over its centre these results are proved in 11 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of open image in new window to k which is as null as possible will be called null the simple open image in new window modules with a null central character are called supersingular when g gl n we show that each simple open image in new window module of dimension n containing a character of the affine subring open image in new window is supersingular using the minimal expressions of haines generalized to open image in new window and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 10 and when n 3 rachel ollivier

the affineq schur algebra abstract we introduce an analogue of the q schur algebra associated to coxeter systems of type a n 1 we give two constructions of this algebra the first construction realizes the algebra as a certain endomorphism algebra arising from an affine hecke algebra of type a r 1 where n r this generalizes the original q schur algebra as defined by dipper and james and the new algebra contains the ordinary q schur algebra and the affine hecke algebra as subalgebras using this approach we can prove a double centralizer property the second construction realizes the affine q schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module analogous to the construction of the ordinary q schur algebra as a quotient of u g l n