integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

cluster concealed algebras abstract the cluster tilted algebras have been introduced by buan marsh and reiten they are the endomorphism rings of cluster tilting objects t in cluster categories we call such an algebra cluster concealed in case t is obtained from a preprojective tilting module for example all representation finite cluster tilted algebras are cluster concealed if c is a representation finite cluster tilted algebra then the indecomposable c modules are shown to be determined by their dimension vectors for a general cluster tilted algebra c we are going to describe the dimension vectors of the indecomposable c modules in terms of the root system of a quadratic form the roots may have both positive and negative coordinates and we have to take absolute values

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

homology computations for complex braid groups ii abstract we complete the computation of the integral homology of the generalized braid group b associated to an arbitrary irreducible complex reflection group w of exceptional type in order to do this we explicitly computed the recursively defined differential of a resolution of z as a z b module using parallel computing we also deduce from this general computation the rational homology of the milnor fiber of the singularity attached to most of these reflection groups

tame wild dichotomy of birkhoff type problems for nilpotent linear operators abstract inspired by recent results of ringel schmidmeier kussin lenzing meltzer xiao wu chen and pu zhang given a field k m 1 and a finite poset i i with a unique maximal element we study the category m o n i f m of mono representations of i over the frobenius k algebra f m k t t m of k dimension m viewed as k vector spaces u with an m nilpotent k linear operator t u u together with t invariant subspaces u i u j u for all i j in i the problem of when the krull schmidt k category m o n i f m is of wild resp tame representation type is called a wild resp tame birkhoff type problem for m nilpotent operators in case when k is algebraically closed we give a complete solution of the problem by describing all minimal pairs i m resp all pairs with m 1 such that category m o n i f m is of wild resp tame representation type we reduce the problem to a birkhoff type problem for the category fspr i f m m o n i f m of subprojective representations over f m of a larger poset i i the tame wild dichotomy for the category m o n i f m is also proved surprisingly in case when i i a b is the union of two incomparable chains i and i of length i a 1 1 and i b 1 1 with i i the problem is equivalent with the wildness resp tameness of the category coh x p of coherent sheaves over the weighted projective line x p for the weight triple p a b m with a b m 2 studied by kussin lenzing and meltzer 19 in relation with the hypersurface singularity f x 1 a x 2 b x 3 m

garside families and garside germs garside families have recently emerged as a relevant context for extending results involving garside monoids and groups which themselves extend the classical theory of generalized braid groups here we establish various characterizations of garside families that is equivalently various criteria for establishing the existence of normal decompositions of a certain type

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

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

cohomology of deligne lusztig varieties for short length regular elements in exceptional groups abstract we determine the cohomology of deligne lusztig varieties associated with some short length regular elements for split groups of type f 4 and e n as a byproduct we obtain conjectural brauer trees for the principal 14 block of e 7 and the principal 24 block of e 8

degenerations in a class of matrix varieties and prinjective modules abstract let k be an algebraically closed field and let i be a finite partially ordered set to any vector v n i we associate the irreducible variety mat v i with an action of the algebraic group g v i see section 6 the main aim of this paper is to give a description of degenerations of orbits of this action in terms of homomorphisms and extensions of prinjective ki modules

uniqueness of roots up to conjugacy for some affine and finite type artin groups let g be one of the artin groups of finite type b n c n and affine type tilde mathbf a n 1 tilde mathbf c n 1 in this paper we show that if and are elements of g such that k k for some nonzero integer k then and are conjugate in g for the artin group of type a n this was recently proved by gonzalez meneses in fact we prove a stronger theorem from which the above result follows easily by using descriptions of those artin groups as subgroups of the braid group on n 1 strands let p be a subset of 1 n an n braid is said to be p pure if its induced permutation fixes each i in p and p straight if it is p pure and it becomes trivial when we delete all the ith strands for i not in p exploiting the nielsen thurston classification of braids we show that if and are p pure n braids such that k k for some nonzero integer k then there exists a p straight n braid with 1 moreover if 1 in p the conjugating element can be chosen to have the first strand algebraically unlinked with the other strands especially in case of p 1 n our result implies the uniqueness of roots of pure braids which was known by bardakov and by kim and rolfsen

linear representations symmetric products and the commuting scheme abstract we show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices as a consequence we give a surjection from the ring of invariants of several matrices to the ring of multisymmetric functions generalizing a classical result of h weyl and f junker we also find a surjection from the ring of invariants over the commuting scheme to the ring of multisymmetric functions this surjection is an isomorphism over a characteristic zero field and induces an isomorphism at the level of reduced structures over an infinite field of positive characteristic

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

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

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

a peak reduction functor for socle projective representations throughout this paper k is a fixed algebraically closed field given a finite poset i e partially ordered set j with the partial order u we denote by max j the set of all maximal elements of j with respect to the order u and we set jys j max j the poset j is said to be an r peak poset provided max j s r throughout this paper we suppose that j s 4 4 1 n p p and max j s p p without loss of generality 1 r 1 r we can assume that the partial order u in j is such that i u j implies 4 i f j in the natural order for all i j g 1 n w x we recall from 18 that the category j spr of peak j spaces or socle projective representations of j over the field k is defined as follows the objects of j spr are systems m s m of finite dimensional k vector j jg j spaces m such that j

eigenspace arrangements of reflection groups the lattice of intersections of reflecting hyperplanes of a complex reflection group w may be considered as the poset of 1 eigenspaces of the elements of w in this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset after posing the main question of whether this poset is shellable we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family g m p n of complex reflection groups and the first 31 of the 34 exceptional groups by constructing cl shellings in addition we completely determine when these eigenspaces of w form a k 1 resp free arrangement for the symmetric group we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets presented as particular upper order ideals of dowling lattices study the representation afforded by the top co homology group and give a simple map to the posets of pointed d divisible partitions 1 main question and results let v denote an n dimensional c vector space a reflection in v is any nonidentity element r in gl v of finite order that fixes some hyperplane hr and a finite subgroup w of gl v is called a reflection group if it is generated by reflections example the action of the symmetric group sn on n 1 2 n gives rise to a faithful action on c via ei e i where e1 e2 en denote the standard basis vectors since the transpositions act as reflections and generate the group this representation realizes sn as a reflection group in gl c we shall refer to it as the defining representation for an element g w and root of unity let v g denote the eigenspace of g in v define e w to be the w poset partially ordered set of all such eigenspaces v g g w ordered by reverse inclusion with w action given by h v g v hgh the eigenvalue 1 gives the lattice lw of intersections of reflecting hyperplanes for w see 29 lemma 4 4 the minimal elements of e w i e the inclusion maximal eigenspaces forw are the focus of springer s theory of regular elements 35 and each has dimension equal to the number a d of degrees d1 d2 dn of w that are divisible by d the order of see proposition 3 7 below when w is crystallographic the poset e w itself appears in broue malle and michel s sylow theory 8 2010 mathematics subject classification primary 20f55 05e10 05e18 55p20 14n20

a reduced tits quadratic form and tameness of three partite subamalgams of tiled orders let d be a complete discrete valuation domain with the unique maximal ideal p we suppose that d is an algebra over an algebraically closed field k and d p k subamalgam d suborders of a tiled d order are studied in the paper by means of the integral tits quadratic form q zn1 2n3 2 z a criterion for a subamalgam d order to be of tame lattice type is given in terms of the tits quadratic form q and a forbidden list 1 17 of minor d suborders of presented in the tables

the bmm symmetrising trace conjecture for the exceptional 2 reflection groups of rank 2 abstract we prove the symmetrising trace conjecture of broue malle and michel for the generic hecke algebra associated to the exceptional irreducible complex reflection group g 13 our result completes the proof of the conjecture for the exceptional 2 reflection groups of rank 2

a narrow over ring adjustment functor such that xi is finitely generated xl is finitely cogenerated 9 is surjective and the a homomorphism cp xl hom an xg adjoint to rp is injective the category of adjusted right r modules will be denoted by adj r note that if r is an artin algebra then x is in adj r if and only if x is finitely generated sot x is a b module and top x is an a module via the natural epimorphisms a t r b in this case adj i r is one of the classes of modules considered by auslander and smal l and therefore adj has the auslander reiten sequences 1 proposition 6 51 the category adjg r was introduced in 25 26 in order to give a module theoretical interpretation of arbitrary bimodule matrix problems in the sense of drozd and roiter it was shown in 26 that if n kop x k mod k is an additive k functor k is a field k and l are additive krull schmidt k categories and mat jv is the category of matrices then there are a faithfully right b traced ring r and a full dense functor 0 mat wl adj r preserving the representation type and such that 0 vanishes only on a small set of trivial idecomposable matrices see also 14 this allows to show in 14 261 that the category mat n has auslander reiten sequences see also 30 moreover a coxeter map is defined in terms of 0 which provides us a useful computa

freeness of multi reflection arrangements via primitive vector fields abstract in 2002 terao showed that every reflection multi arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations we first generalize terao s result to multi arrangements stemming from well generated unitary reflection groups where the multiplicity of a hyperplane depends on the order of its stabilizer here the exponents depend on the exponents of the dual reflection representation we then extend our results further to all imprimitive irreducible unitary reflection groups in this case the exponents turn out to depend on the exponents of a certain galois twist of the dual reflection representation that comes from a beynon lusztig type semi palindromicity of the fake degrees

non crossing partition lattices in finite real reflection groups for a finite real reflection group w with coxeter element we give a case free proof that the closed interval i forms a lattice in the partial order on w induced by reflection length key to this is the construction of an isomorphic lattice of spherical simplicial complexes we also prove that the greatest element in this latter lattice embeds in the type w simplicial generalised associahedron and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere

finite complex reflection arrangements are k pi 1 let v be a finite dimensional complex vector space and w gl v be a finite complex reflection group let v reg be the complement in v of the reflecting hyperplanes we prove that v reg is a k 1 space this was predicted by a classical conjecture originally stated by brieskorn for complexified real reflection groups the complexified real case follows from a theorem of deligne and after contributions by nakamura and orliksolomon only six exceptional cases remained open in addition to solving these six cases our approach is applicable to most previously known cases including complexified real groups for which we obtain a new proof based on new geometric objects we also address a number of questions about 1 w v the braid group of w this includes a description of periodic elements in terms of a braid analog of springer s theory of regular elements

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

a class of garside groupoid structures on the pure braid group we construct a class of garside groupoid structures on the pure braid groups one for each function called labelling from the punctures to the integers greater than 1 the object set of the groupoid is the set of ball decompositions of the punctured disk the labels are the perimeters of the regions our construction generalises garside s original garside structure but not the one by birman ko lee as a consequence we generalise the tamari lattice ordering on the set of vertices of the associahedron

on borcherds type lie algebras for each even lattice mathcal l there is a canonical way to construct an infinite dimensional lie algebra via lattice vertex operator algebra theory we call this lie algebra and its subalgebras the borcherds type lie algebras associated to mathcal l in this paper we apply this construction to even lattices arising from representation theory of finite dimensional associative algebras this is motivated by the different realizations of kac moody algebras by borcherds using lattice vertex operators and by peng xiao using ringel hall algebras respectively for any finite dimensional algebra a of finite global dimension we associate a borcherds type lie algebra mathfrak bl a to a in contrast to the ringel hall lie algebra approach mathfrak bl a only depends on the symmetric euler form or tits form but not the full representation theory of a however our results show that for certain classes of finite dimensional algebras whose representation theory is controlled by the euler bilinear forms or tits forms their borcherds type lie algebras do have close relations with the representation theory of these algebras beyond the class of hereditary algebras these algebras include canonical algebras representation directed algebras and incidence algebras of finite prinjective types

on weakly non negative unit forms and tame algebras in this paper we consider integral quadratic forms q ip z of the form q x 2i x i jqijx xj and call them unit forms such a unit form q is said to be weakly non negative if q x q whenever jt2 0 that is whenever x o for i 1 n our main purpose is to formulate a criterion for weak non negativity of unit forms this question arises in various areas of representation theory in connection with the characterization of tame representation type in the representation theory of quivers 17 6 partially ordered sets 18 19 certain classes of bimodules 8 23 1 3 and certain classes of algebras 20 14 24 16 for each of these structures q there is a unit form qn naturally attached to q such that the weak non negativity of qn is an equivalent or at least a necessary condition for q to be of tame representation type a class of algebras for which tame representation type is equivalent to the weak non negativity of the associated unit form is described in the last section of this paper a general criterion for weak non negativity using determinants was given by zeldich 28 see also 21 another criterion was obtained by de la pena 21 our approach is similar to that in 11 where we gave a characterization of weakly positive unit forms based on the classification of the critical forms let us recall some notions given a unit form q ip tl and a non empty subset ix im of 1 n we denote by qt the quadratic form qd where d z it maps the cth natural basis vector ek onto eik such a unit form q is called the restriction of q to we call a unit form q on ip a hypercritical form if q is weakly non negative for every proper subset 1 n though q itself is not in these terms a unit form q is weakly non negative if and only if q admits no hypercritical restriction so we have to classify the hypercritical forms in 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of fig 1 on p 50 in particular there are only finitely many hypercritical forms however their number is large by a simple process we reduce the problem to the classification of the stretched hypercritical forms in 2 we show that except for a few of them these are of the shape q x q x xn xn ixn where q q n d is critical in a similar way the unit forms attached to the hyperbolic graphs am dm em correspond to those

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

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

the existence of hall polynomials for posets of finite prinjective type abstract we prove the existence of hall polynomials for prinjective representations of finite partially ordered sets of finite prinjective type in section 4 we shortly discuss consequences of the existence of hall polynomials in particular we are able to define a generic ringel hall algebra for prinjective representations of posets of finite prinjective type

completions branched covers artin groups and singularity theory we study the curvature of metric spaces and branched covers of riemannian manifolds with applications in topology and algebraic geometry here curvature bounds are expressed in terms of the cat inequality we prove a general cat extension theorem giving sufficient conditions on and near the boundary of a locally cat metric space for the completion to be cat we use this to prove that a branched cover of a complete riemannian manifold is locally cat if and only if all tangent spaces are cat 0 and the base has sectional curvature bounded above by we also show that the branched cover is a geodesic space using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally cat and the complement of the branch locus to be contractible we conjecture that the universal branched cover of cn over the mirrors of a finite coxeter group is cat 0 conditionally on this conjecture we use our machinery to prove the arnoldpham thom conjecture on k 1 spaces for artin groups also conditionally we prove the asphericity of moduli spaces of amply lattice polarized k3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in arnold s hierar

normalisateur et groupe d artin de type spherique abstract let a s be an artin group and x a subset of s denote by ax the subgroup of a generated by x when a is of spherical type we prove that the normalizer and the commensurator of ax in a are equal and are the product of ax by the quasi centralizer of ax in a looking the associated monoids a and ax we described the quasi centralizer of ax in a thanks to results in coxeter groups these two results generalize earlier results of paris j algebra 196 1997 369 399 finally we compare in the spherical case the normalizer of a parabolic subgroup in the artin group and in the coxeter group

tame prinjective type and tits form of two peak posets ii throughout this paper we use the terminology and notation introduced w x in 5 7 11 13 in particular k is an algebraically closed field and we denote by i u a finite poset i e partially ordered set with respect to the partial order u we shall write i j if i u j and i j for the sake of simplicity we write i instead of i u we denote by max i the set of all maximal elements of i and i will be called an n peak poset if max i s n let ki be the incidence k algebra of the poset i that is ki is the k subalgebra of the full i i matrix algebra m k consisting of all i w x matrices l in m k such that l s 0 if i uu j in i u we recall from i j i i j w x 13 that the socle soc ki of the right ki module ki is projective and the global dimension of the algebra ki is finite we denote by mod ki the category of finitely generated right ki mod ules and by prin ki the full subcategory of mod ki consisting of all prinjective modules that is modules x in mod ki having a projective resolution of the form

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

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

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

gaussian groups and garside groups two generalisations of artin groups it is known that a number of algebraic properties of the braid groups extend to arbitrary finite coxeter type artin groups here we show how to extend the results to more general groups that we call garside groups define a gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths and where left and right lowest common multiples exist a garside monoid is a gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition a gaussian group is the group of fractions of a gaussian monoid and a garside group is the group of fractions of a garside monoid braid groups and more generally finite coxeter type artin groups are garside groups we determine algorithmic criteria in terms of presentations for recognizing gaussian and garside monoids and groups and exhibit infinite families of such groups we describe simple algorithms that solve the word problem in a gaussian group show that these algorithms have a quadratic complexity if the group is a garside group and prove that garside groups have quadratic isoperimetric inequalities we construct normal forms for gaussian groups and prove that in the case of a garside group the language of normal forms is regular symmetric and geodesic has the 5 fellow traveller property and has the uniqueness property this shows in particular that garside groups are geodesically fully biautomatic finally we consider an automorphism of a finite coxeter type artin group derived from an automorphism of its defining coxeter graph and prove that the subgroup of elements fixed by this automorphism is also a finite coxeter type artin group that can be explicitly determined 1991 mathematics subject classification primary 20f05 20f36 secondary 20b40 20m05

bipartite posets of finite prinjective type one of the main results of this paper is theorem 1 2 which contains a characterization of finite bipartite posetsi i i for which the category prin ki of prinjective modules over the incidencek algebrakiofiis of finite representation type wherekis a field in particular it is shown that for any such poseti the auslander reiten quiver of the category prin ki has no oriented cycle and there is a bijection between the isomorphism classes of indecomposable objects in prin ki and the positive roots of the quadratic formqiassociated withi an existence of a preprojective component in prin ki is proved for faithful posetsiwhich area free

generalized symmetric functions and invariants of matrices we generalize the classical isomorphism between symmetric functions and invariants of a matrix in particular we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra the main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra

on non conjugate coxeter elements in well generated reflection groups given an irreducible well generated complex reflection group w with coxeter number h we call a coxeter element any regular element in the sense of springer of order h in w this is a slight extension of the most common notion of coxeter element we show that the class of these coxeter elements forms a single orbit in w under the action of reflection automorphisms for coxeter and shephard groups this implies that an element c is a coxeter element if and only if there exists a simple system s of reflections such that c is the product of the generators in s we moreover deduce multiple further implications of this property in particular we obtain that all noncrossing partition lattices of w associated to different coxeter elements are isomorphic we also prove that there is a simply transitive action of the galois group of the field of definition of w on the set of conjugacy classes of coxeter elements finally we extend several of these properties to springer s regular elements of arbitrary order

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

quadratic bimodules and quadratic orders in some branches of mathematics especially the representation theory of artin algebras and orders it has been an important problem to study the geometric quotient gln mn m for a module m g m gmg 1 for g gln and m mn m an effective formulation of such problems is a bimodule problem 4 9 which is a study of the matrix category mat c m associated to an additive category c and a c c module m section 2 1 the bimodule problems have nice module theoretical interpretation by means of prinjective modules 22 29 30 they have already many useful applications in the representation theory of finite dimensional algebras artinian rings orders vector space categories and bocses see 6 7 9 16 19 25 28 29 in this paper we introduce the concept of quadratic extensions of rings section 1 2 which are closely related to many rings known in representation theory for example bass orders 10 17 backstrom orders 28 green orders 27 special biserial algebras 31 and clannish algebras 5 see sections 1 3 3 4 and 3 5 using quadratic extensions we introduce the concept of quadratic bimodules section 3 1 where we can easily formulate self reproducing systems 21 and clans 5 in terms of quadratic bimodules then we study their matrix category mat c m associated to a quadratic c c module m especially we give a description of all indecomposable objects in terms of walks sections 3 3 6 1 which can be regarded as a generalization of green walks 13 section 3 4 strings and bands 5 31 section 3 5 as an application we give a description of finitely generated indecomposable modules over quadratic orders section 1 3 2 in section 7 3 in forthcoming papers our results will be applied in construction of large new classes of orders of tame representation type some of the results of this paper were presented at the conference and workshop constanta 2000 and were announced in 19 without proof

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1