an effective lie kolchin theorem for quasi unipotent matrices abstract we establish an effective version of the classical lie kolchin theorem namely let a b gl m c be quasi unipotent matrices such that the jordan canonical form of b consists of a single block and suppose that for all k 0 the matrix a b k is also quasi unipotent then a and b have a common eigenvector in particular a b gl m c is a solvable subgroup we give applications of this result to the representation theory of mapping class groups of orientable surfaces

homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

commutativity preserving linear maps and lie automorphisms of triangular matrix algebras abstract in this article we classify linear maps from the algebra t n of n upper triangular matrices into itself satisfying ab ba 0 if and only if ab ba 0 in particular we show that for n 2 any such map is either the sum of an algebra automorphism and a map into the centre or the sum of the negative of an algebra anti automorphism and a map into the centre as a consequence lie automorphisms of the algebra t n are also classified

zariski theorems and diagrams for braid groups empirical properties of generating systems for complex reflection groups and their braid groups have been observed by orlik solomon and broue malle rouquier using shephard todd classification we give a general existence result for presentations of braid groups which partially explains and generalizes the known empirical properties our approach is invariant theoretic and does not use the classification the two ingredients are springer theory of regular elements and a zariski like theorem

a homological representation formula of colored alexander invariants we give a formula of the colored alexander invariant in terms of the homological representation of the braid groups which we call truncated lawrence s representation this formula generalizes the famous burau representation formula of the alexander polynomial

parameters for which the lawrence krammer representation is reducible we show that the representation introduced by lawrence and krammer to show the linearity of the braid group is generically irreducible however for some values of its two parameters when these are specialized to complex numbers it becomes reducible we construct a representation of degree n n 1 2 of the bmw algebra of type a n 1 as a representation of the braid group on n strands it is equivalent to the lawrence krammer representation where the two parameters of the bmw algebra are related to those appearing in the lawrence krammer representation we give the values of the parameters for which the representation is reducible and give the proper invariant subspaces in some cases we use this representation to show that for these special values of the parameters the bmw algebra of type a n 1 is not semisimple

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

homological representations of the hecke algebra in this paper a topological construction of representations of thean 1 series of hecke algebras associated with 2 row young diagrams will be given this construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection the fibres of the vector bundle are homology spaces of configuration spaces of points in c with a suitable twisted local coefficient system it is also shown that there is a close correspondence between this construction and the work of tsuchiya and kanie who constructed hecke algebra representations from the monodromy ofn point functions in a conformal field theory onp1 this work has significance in relation to the one variable jones polynomial which can be expressed in terms of characters of the iwahori hecke algebras associated with 2 row young diagrams it gives rise to a topological description of the jones polynomial which will be discussed elsewhere l2

the burau representation of the braid group b_n is unfaithful for large n two fundamental theorems of classical knot theory by j alexander are that every knot is a closed braid and second that a certain procedure see 1 2 for both results assigns to each knot k a polynomial a k in t which only depends on the topological type of the knot once normalized by a power of t to take a positive value at t 0 with this normalization take t to be a transcendental number since 1926 2 it has been known that the function a knots c is not one to one however it has not become clear whether the composite

on the category o for rational cherednik algebras we study the category o of representations of the rational cherednik algebra aw attached to a complex reflection group w w e construct an exact functor called knizhnik zamolodchikov functor o hw mod where hw is the finite iwahori hecke algebra associated to w we prove that the knizhnik zamolodchikov functor induces an equiv alence between o otor the quotient of o by the subcategory of aw modules supported on the discriminant and the category of finite dimensional hw modules the standard aw modules go under this equivalence to certain modules arising in kazhdan lusztig theory of cells provided w is a weyl group and the hecke algebra hw has equal parameters we prove that the category o is equivalent to the module category over a finite dimensional algebra a generalized q schur algebra associated to w

divisorial inertia and central elements in braid groups abstract given a complex reflection group w we will show how the generators of the centers of the parabolic subgroups of the pure braid group p w can be represented by loops around irreducible divisors of the corresponding minimal de concini procesi model x w we will also show that a more subtle construction gives representations of the generators of the centers of the parabolic subgroups of the braid group b w as loops in the not smooth quotient variety x w w

commuting centralizing traces and lie triple isomorphisms on triangular algebras revisited abstract the aim of the paper is to give a description of commuting centralizing traces of bilinear maps and lie triple isomorphisms on certain triangular algebras as consequences commuting centralizing traces of bilinear maps and lie triple isomorphisms on upper triangular matrix algebras over a 2 torsion free commutative ring are completely determined

flat connections and brauer type algebras in this paper we introduce a brauer type algebra bg associated with every pseudo reflection group and every coxeter group g when g is a coxeter group of simply laced type we show that bg is isomorphic to the generalized brauer algebra of simply laced type introduced by cohen gijsbers and wales 2005 cgw1 we also prove that bg has a cellular structure and be semisimple for generic parameters when g is a dihedral group or the type h3 coxeter group moreover in the process of construction we introduce a further generalization of the lawrence krammer representation to complex braid groups associated with all pseudo reflection groups

quasiharmonic polynomials for coxeter groups and representations of cherednik algebras we introduce and study deformations of finite dimensional modules over rational cherednik algebras our main tool is a generalization of usual harmonic polynomials for each coxeter group the so called quasiharmonic polynomials a surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all coxeter groups

on twisted lawrence krammer representations abstract lawrence krammer representations are an important family of linear representations of artin tits groups of small type which are known under some assumptions on the parameters to be faithful when the type is spherical or more generally when they are restricted to the artin tits monoid and irreducible when the type is connected here we investigate an analogue of these representations introduced by digne in the spherical cases for every artin tits monoid that appears as the submonoid of fixed points of an artin tits monoid of small type under a group of graph automorphisms and for the corresponding artin tits group under the same assumptions on the parameters as in the small type cases we first show that these so called twisted lawrence krammer representations are faithful and we then prove by computing their formulas when the group of graph automorphisms is of order two or three their irreducibility in all the spherical and connected cases but one

commuting traces and commutativity preserving maps on triangular algebras abstract let a be a triangular algebra the problem of describing the form of a bilinear map b a a a satisfying b x x x x b x x for all x a is considered as an application commutativity preserving maps and lie isomorphisms of certain triangular algebras e g upper triangular matrix algebras and nest algebras are determined

spectral properties of the canonical solution operator to we study boundedness compactness and schatten class membership of the canonical solution operator to restricted to 0 1 forms with holomorphic coefficients on l2 d where is a measure with the property that the monomials form an orthogonal family in l2 d the characterizations are formulated in terms of moment properties of our results generalize the results of the first author to several variables contain some known results for several variables and also cover new ground

commuting traces of biadditive mappings commutativity preserving mappings and lie mappings biadditive mappings b r r r where r is a prime ring with certain additional properties satisfying b x x x xb x x for all x r are characterized as an application we determine the structures of commutativity preserving mappings lie isomorphisms and lie derivations of certain prime rings

formula for the reflection length of elements in the group g m p n we deduce a closed formula for the reflection length functions on the reflection group g m p n with p m and p m the formula is shown separately in three cases p 1 p m and p 2 m 1 see the proof of theorems 2 1 3 1 and 4 4

braid groups and the co hopfian property abstract let b n be the braid group on n 4 strands we prove that b n modulo its center is co hopfian we then show that any injective endomorphism of b n is geometric in the sense that it is induced by a homeomorphism of a punctured disk we further prove that any injection from b n to b n 1 is geometric for n 7 additionally we obtain analogous results for mapping class groups of punctured spheres the methods use thurston s theory of surface homeomorphisms and build upon work of ivanov mccarthy

the sign representation for shephard groups shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes for a shephard group we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes this representation turns out to have many equivalent guises making it analogous to the sign representation of a finite coxeter group one of these guises is up to a twist the cohomology of the milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invariant of minimal degree

torsion subgroups of quasi abelianized braid groups abstract this article deals with braid groups of complex reflection groups as introduced in 2 we present results in two directions first we extend the works of goncalves guaschi ocampo 5 and marin 7 on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group we get explicit criteria for subgroups of the complex reflection group to lift to subgroups of this quotient in the specific case of the classical braid group this enables us to describe all its finite subgroups we show that every odd order finite group can be embedded in it when the number of strands goes to infinity we also determine a complete list of the irreducible reflection groups for which this quotient is a bieberbach group in the second part we describe the abelianization of the inverse image of a subgroup of a reflection group in its generalized braid group generalizing results of the first author 1 in particular these abelianized groups are not necessarily torsion free

bmw algebras of simply laced type abstract it is known that the recently discovered representations of the artin groups of type a n the braid groups can be constructed via bmw algebras we introduce similar algebras of type d n and e n which also lead to the newly found faithful representations of the artin groups of the corresponding types we establish finite dimensionality of these algebras moreover they have ideals i 1 and i 2 with i 2 i 1 such that the quotient with respect to i 1 is the hecke algebra and i 1 i 2 is a module for the corresponding artin group generalizing the lawrence krammer representation finally we give conjectures on the structure the dimension and parabolic subalgebras of the bmw algebra as well as on a generalization of deformations to brauer algebras for simply laced spherical type other than a n

k commuting maps on triangular algebras abstract in this paper k commuting maps on certain triangular algebras are determined as an application we show that every k commuting map on an upper triangular matrix algebra over a unital commutative ring of 2 torsion free or a nest algebra is proper

the braid group b4 is linear we study a linear representation b n gl m z q 1 t 1 with m n n 1 2 we will show that for n 4 this representation is faithful we prove a relation with the new charney length function we formulate a conjecture implying that is faithful for all n

configurations braids and homotopy groups the main results of this article are certain connections between braid groups and the homotopy groups of the 2 sphere the connections are given in terms of brunnian braids over the disk and over the 2 sphere the techniques arise from the natural structure of simplicial and structures on fundamental groups of configuration spaces

hankel operators and the stieltjes moment problem let s be a non vanishing stieltjes moment sequence and let be a representing measure of it we denote by n the image measure in cn of n under the map t t where n is the rotation invariant probability measure on the unit sphere we show that the closure of holomorphic polynomials in l2 n is a reproducing kernel hilbert space of analytic functions and describe various spectral properties of the corresponding hankel operators with anti holomorphic symbols in particular if n 1 we prove that there are nontrivial hilbert schmidt hankel operators with anti holomorphic symbols if and only if s is exponentially bounded in this case the space of symbols of such operators is shown to be the classical dirichlet space we mention that the classical weighted bergman spaces the hardy space and fock type spaces fall in this setting

non crossing partitions of type e e r abstract we investigate a new lattice of generalised non crossing partitions constructed using the geometry of the complex reflection group g e e r for the particular case e 2 resp r 2 our lattice coincides with the lattice of simple elements for the type d n resp i 2 e dual braid monoid using this lattice we construct a garside structure for the braid group b e e r as a corollary one may solve the word and conjugacy problems in this group

braid groups are linear in a previous work 11 the author considered a representation of the braid group p bn glm z ql1 t l m n n 1 2 and proved it to be faithful for n 4 bigelow 3 then proved the same representation to be faithful for all n by a beautiful topological argument the present paper gives a different proof of the faithfulness for all n we establish a relation between the charney length in the braid group and exponents of t a certain bn invariant subset of the module is constructed whose properties resemble those of convex cones we relate line segments in this set with the thurston normal form of a braid

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

functional identities of degree 2 in triangular rings abstract let r be a triangular ring the problem of describing the form of additive maps f 1 f 2 g 1 g 2 r r satisfying functional identity f 1 x y f 2 y x xg 2 y yg 1 x 0 for all x y r is considered as an application generalized inner biderivations and commuting additive maps of certain triangular rings are determined

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

some spectral properties of the canonical solution operator to on weighted fock spaces abstract we characterize the schatten class membership of the canonical solution operator to acting on l 2 e 2 where is a subharmonic function with a doubling measure the obtained characterization is in terms of as part of our approach we study hankel operators with anti analytic symbols acting on the corresponding fock space of entire functions in l 2 e 2

the lawrence krammer bigelow representations of the braid groups via uq sl2 abstract we construct representations of the braid groups b n on n strands on free z q 1 s 1 modules w n l using generic verma modules for an integral version of u q sl 2 we prove that the w n 2 are isomorphic to the faithful lawrence krammer bigelow representations of b n after appropriate identification of parameters of laurent polynomial rings by constructing explicit integral bases and isomorphism we also prove that the b n representations w n l are irreducible over the fractional field q q s

continuity and schatten von neumann p class membership of hankel operators with anti holomorphic symbols on generalized fock spaces abstract in this paper we investigate hankel operators h f a m 2 a m 2 with anti holomorphic symbols f k 0 b k z k l 2 c z m where a m 2 are general fock spaces we will show that h f is not continuous if the corresponding symbol is not a polynomial f k 0 n b k z k for polynomial symbols we will give necessary and sufficient conditions for continuity and compactness in terms of n and m for monomials z k we will give a complete characterization of the schatten von neumann p class membership for p 0 namely in case 2 k m the hankel operators h z k are in the schatten von neumann p class iff p 2 m m 2 k and in case 2 k m they are not in the schatten von neumann p class

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

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

on injective homomorphisms for pure braid groups and associated lie algebras abstract the purpose of this article is to record the center of the lie algebra obtained from the descending central series of artin s pure braid group a lie algebra analyzed in work of kohno t kohno linear representations of braid groups and classical yang baxter equations in contemp math vol 78 1988 pp 339 363 t kohno vassiliev invariants and the de rham complex on the space of knots in symplectic geometry and quantization in contemp math vol 179 amer math soc providence ri 1994 pp 123 138 t kohno serie de poincare koszul associee aux groupes de tresses pures invent math 82 1985 57 75 and falk and randell m falk r randell the lower central series of a fiber type arrangement invent math 82 1985 77 88 the structure of this center gives a lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful f r cohen j wu on braid groups free groups and the loop space of the 2 sphere in algebraic topology categorical decomposition techniques in progr math vol 215 birkhauser basel 2003 braid groups free groups and the loop space of the 2 sphere math at 0409307 however it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of p n such as the gassner representation

lie isomorphisms of nest algebras abstract in this paper we characterize linear maps between two nest algebras t n and t m which satisfy the property that ab ba a b b a for all a b t n in particular it is shown that such isomorphisms only exist if n is similar to m or m

residual finiteness growths of virtually special groups let g be a virtually special group then the residual finiteness growth of g is at most linear this result cannot be found by embedding g into a special linear group indeed the special linear group mathrm sl k mathbb z for k 2 has residual finiteness growth n k 1

a dual braid monoid for the free group abstract we construct a quasi garside monoid structure for the free group this monoid should be thought of as a dual braid monoid for the free group generalising the constructions by birman ko lee and by the author of new garside monoids for artin groups of spherical type

morphisms from hecke algebras to graphic temperley lieb algebras abstract in this paper we construct many surjective morphisms from the hecke algebras h of uniform type infinite type mostly to certain graphic temperley lieb algebras which provide a way to obtain representations of those hecke algebras when is the coxeter matrix i 2 2 n 1 of the dihedral group d 2 n 1 we show there are altogether n such morphisms exactly realize all n different two dimensional irreducible representations of h in the study the chebyshev polynomials appear and play a prominent role in the computation

multiplicative lie n derivations of triangular rings abstract we introduce the notion of a multiplicative lie n derivation of a ring generalizing the notion of a lie triple derivation the main goal of the paper is to consider the question of when do all multiplicative lie n derivations of a triangular ring t have the so called standard form the main result is applied to the classical examples of triangular rings nest algebras and block upper triangular matrix rings

on lawrence krammer representations lawrence krammer representations or simply lk representations for short are linear representations of artin tits groups of small type over the ring z x 1 y 1 which are of particular interest since they are known to be faithful when the type is spherical and irreducible when the type is spherical and connected here we define and study lk representations over an arbitrary commutative ring r under some conditions on r and on the parameters of an lk representation which generalize the classical settings we recover the faithfulness property and give a new proof of irreducibility that works for every non necessarily spherical connected type we then study the equivalence of distinct lk representations of a given artin tits group for distinct choices of parameters finally we generalize to our setting the known classification of lk representations in the spherical cases and proceed with their classification in the affine cases

reading the dual garside length of braids from homological and quantum representations we show that lawrence s representation and linear representations of the braid groups from generic highest weight representations of u q mathfrak sl 2 detect the dual garside length of braids in a simple and natural way that is by expressing a representation as a matrix over a laurent polynomial ring using a certain natural basis the span of the variable is equal to a constant multiple of the dual garside length

universal deformations of the finite quotients of the braid group on 3 strands abstract we prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively have finite rank this is a special case of a conjecture by broue malle and rouquier for the generic hecke algebra of an arbitrary complex reflection group exploring the consequences of this case we prove that we can determine completely the irreducible representations of this braid group of dimension at most 5 thus recovering a classification of tuba and wenzl in a more general framework

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

krammer representations for complex braid groups abstract let b be the generalized braid group associated to some finite complex reflection group w we define a representation of b of dimension the number of reflections of the corresponding reflection group which generalizes the krammer representation of the classical braid groups and is thus a good candidate in view of proving the linearity of b we decompose this representation in irreducible components and compute its zariski closure as well as its restriction to parabolic subgroups we prove that it is faithful when w is a coxeter group of type ade and odd dihedral types and conjecture its faithfulness when w has a single class of reflections if true this conjecture would imply various group theoretic properties for these groups that we prove separately to be true for the other groups

truncations and extensions of the brauer chen algebra abstract the brauer chen algebra is a generalization of the algebra of brauer diagrams to arbitrary complex reflection groups that admits a natural monodromic deformation we determine the generic representation theory of the first non trivial quotient of this algebra we also define natural extensions of this algebra and prove that they similarly admit natural monodromic deformations

on coxeter diagrams of complex reflection groups we study complex coxeter diagrams of some unitary reflection groups using solely the combinatorics of diagrams we give a new proof of the classification of root lattices defined over e z e 2 i 3 there are only four such lattices namely the e lattices whose real forms are a2 d4 e6 and e8 next we address the issue of characterizing the diagrams for unitary reflection groups a question that was raised by broue malle and rouquier to this end we describe an algorithm which given a unitary reflection group g picks out a set of complex reflections if g is the reflection group of a root lattice defined over e the reflections selected by the algorithm form the known diagram for g for the reflection group of the complex coxeter todd lattice k e 12 we find a new diagram that extends to an affine diagram withz 7z symmetry if g is a weyl group the algorithm immediately yields a set of simple roots otherwise experimental evidences indicate that the algorithm selects a minimal generating set of reflections if g is primitive and g has a set of roots whose z span is a discrete subset of the ambient vector space

cuspidal calogero moser and lusztig families for coxeter groups the goal of this paper is to compute the cuspidal calogero moser families for all infinite families of finite coxeter groups at all parameters we do this by first computing the symplectic leaves of the associated calogero moser space and then by classifying certain rigid modules numerical evidence suggests that there is a very close relationship between calogero moser families and lusztig families our classification shows that additionally the cuspidal calogero moser families equal cuspidal lusztig families for the infinite families of coxeter groups

on the quasi heredity of birman wenzl algebras abstract in this paper we consider the birman wenzl algebras over an arbitrary field and prove that they are cellular in the sense of graham and lehrer furthermore we determine for which parameters the birman wenzl algebras are quasi hereditary so the general theory of cellular algebras and quasi hereditary algebras applies to birman wenzl algebras as a consequence we can determine all irreducible representations of the birman wenzl algebras by linear algebra methods we prove also that the new hecke algebras induced from birman wenzl algebras are frobenius over a field but not always cellular

monoids in the fundamental groups of the complement of logarithmic free divisors in c3 abstract we study monoids generated by certain zariski van kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in c 3 listed by sekiguchi these monoids admit positive homogeneous presentations theorem 1 five of them are artin monoids and eight of them are free abelian monoids the remaining four monoids are not gausian and hence are neither garside nor artin theorem 2 however we introduce the concept of fundamental elements for positive homogeneously presented monoids and we show that all 17 monoids possess fundamental elements theorem 3 as an application of the study of monoids we solve some decision problems for the fundamental groups except in three cases

on krammer s representation of the braid group a connection is made between the krammer representation and the birman murakami wenzl algebra inspired by a dimension argument a basis is found for a certain irrep of the algebra and relations which generate the matrices are found following a rescaling and change of parameters the matrices are found to be identical to those of the krammer representation the two representations are thus the same proving the irreducibility of one and the faithfulness of the other

finite dimensional representations of the rational cherednik algebra for g4 abstract in this paper we study representations of the rational cherednik algebra associated to the complex reflection group g 4 in particular we classify the irreducible finite dimensional representations and compute their characters

normalisers of parabolic subgroups in finite unitary reflection groups abstract it is well known that the normaliser of a parabolic subgroup of a finite coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots we show that a similar result holds for all finite unitary reflection groups namely the normaliser of a parabolic subgroup is a semidirect product and in many but not all cases the complement can be obtained as the stabiliser of a set of roots in addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup

functional identities of degree 2 in arbitrary triangular rings abstract the aim of the paper is to give the solutions of functional identities of degree 2 in arbitrary triangular rings which generalizes a recent result due to eremita as applications commuting additive maps and generalized inner biderivations of arbitrary triangular rings are determined

automorphisms of certain lie algebras of upper triangular matrices over a commutative ring abstract letrbe an arbitrary commutative ring with identity denote bytthe lie algebra overrconsisting of all upper triangularnbynmatrices overrand letbbe the lie subalgebra oftconsisting of all matrices of trace 0 the aim of this paper is to give an explicit description of the automorphism group of the lie algebrat which extends a result given by dokovic in addition we explicitly describe the automorphism group of the lie algebrabwhennis a unit ofr

analytic discs in the boundary and compactness of hankel operators with essentially bounded symbols abstract we derive conditions for compactness of hankel operators h f a 2 l 2 h f g i p f g with bounded holomorphic symbols f for a large class of convex and bounded domains with d k

braid groups of imprimitive complex reflection groups abstract we obtain new presentations for the imprimitive complex reflection groups of type d e e r and their braid groups b d e e r for d r 2 diagrams for these presentations are proposed the presentations have much in common with coxeter presentations of real reflection groups they are positive and homogeneous and give rise to quasi garside structures diagram automorphisms correspond to group automorphisms the new presentation shows how the braid group b d e e r is a semidirect product of the braid group of affine type a r 1 and an infinite cyclic group elements of b d e e r are visualised as geometric braids on r 1 strings whose first string is pure and whose winding number is a multiple of e we classify periodic elements and show that the roots are unique up to conjugacy and that the braid group b d e e r is strongly translation discrete

automorphisms of the lie algebra of upper triangular matrices over a connected commutative ring abstract let r be a non trivial commutative ring having no idempotents except 0 and 1 denote by t the lie algebra over r consisting of all upper triangular n by n matrices over r we give an explicit description of the automorphism group of this lie algebra

the center of pure complex braid groups abstract broue malle and rouquier 1998 conjectured in 2 that the center of the pure braid group of an irreducible finite complex reflection group is cyclic we prove this conjecture for the remaining exceptional types using the analogous result for the full braid group due to bessis and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center

complete positive group presentations abstract a combinatorial property of positive group presentations called completeness is introduced with an effective criterion for recognizing complete presentations and an iterative method for completing an incomplete presentation we show how to directly read several properties of the associated monoid and group from a complete presentation cancellativity or existence of common multiples in the case of the monoid or isoperimetric inequality in the case of the group in particular we obtain a new criterion for recognizing that a monoid embeds in a group of fractions typical presentations eligible for the current approach are the standard presentations of the artin tits groups and the heisenberg group