a rotation group whose subspace arrangement is not from a real reflection group abstract we exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group answering a question of martino and singh

on the cohen macaulay property of modular invariant rings abstract if v is a faithful module for a finite group g over a field of characteristic p then the ring of invariants need not be cohen macaulay if p divides the order of g in this article the cohomology of g is used to study the question of cohen macaulayness of the invariant ring one of the results is a classification of all groups for which the invariant ring with respect to the regular representation is cohen macaulay moreover it is proved that if p divides the order of g then the ring of vector invariants of sufficiently many copies of v is not cohen macaulay a further result is that if g is a p group and the invariant ring is cohen macaulay then g is a bireflection group i e it is generated by elements which fix a subspace of v of codimension at most 2

the separating variety for the basic representations of the additive group abstract for a group g acting on an affine variety x the separating variety is the closed subvariety of x x encoding which points of x are separated by invariants we concentrate on the indecomposable rational linear representations v n of dimension n 1 of the additive group of a field of characteristic zero and decompose the separating variety into the union of irreducible components we show that if n is odd divisible by four or equal to two the closure of the graph of the action which has dimension n 2 is the only component of the separating variety in the remaining cases there is a second irreducible component of dimension n 1 we conclude that in these cases there are no polynomial separating algebras

invariant fields of finite irreducible reflection groups we prove the following result if g is a finite irreducible reflection group defined over a base field k then the invariant field of g is purely transcendental over k even if g is divisible by the characteristic of k it is well known that in the above situation the invariant ring is in general not a polynomial ring so the question whether at least the invariant field is purely transcendental noether s problem is quite natural

p adic string compactified on a torus u 1 xd model with the villain action on ag loop generalizationfg of the bruhat tits tree for thep adic linear groupgl 2 p is considered all correlation functions and the statistical sum are calculated we compute also the averages of these correlation functions forn vertices attached to the boundary offg when the compactification radius tends to infinity the averages provide theg loopn point amplitudes of the uncompactifiedp adic string theory in particular forg 0 the freund olson amplitudes

on crepant resolution of some hypersurface singularities and a criterion for ufd in this article we find some diagonal hypersurfaces that admit crepant resolutions we also give a criterion for unique factorization domains

finite groups generated in low real codimension abstract we study the intersection lattice of the arrangement a g of subspaces fixed by subgroups of a finite linear group g when g is a reflection group this arrangement is precisely the hyperplane reflection arrangement of g we generalize the notion of finite reflection groups we say that a group g is generated resp strictly generated in codimension k if it is generated by its elements that fix point wise a subspace of codimension at most k resp precisely k we prove that the alternating subgroup alt w of a reflection group w is strictly generated in codimension two further we compute the intersection lattice of all finite subgroups of g l 3 r and moreover we emphasize the groups that are minimally generated in real codimension two i e groups that are strictly generated in codimension two but have no real reflection representations we also provide several examples of groups generated in higher codimension

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

rigidity of graded regular algebras we prove a graded version of alev polo s rigidity theorem the homogenization of the universal enveloping algebra of a semisimple lie algebra and the rees ring of the weyl algebras a n k cannot be isomorphic to their fixed subring under any finite group action we also show the same result for other classes of graded regular algebras including the sklyanin algebras

rings of invariants of finite groups which are hypersurfaces ii let g be a finite subgroup of gl acting naturally on an affine space c of dimension n over the complex number field and denote by v g the quotient variety of c under this action of g an element c of gl is said to be a pseudo reflection if the rank of r 1 is equal to one and g is said to be a reflection group if g is generated by pseudo reflections finite reflection groups were classified by coxeter shephard and todd 8 261 and are characterized by

symmetric powers of modular representations for groups with a sylow subgroup of prime order abstract let v be a representation of a finite group g over a field of characteristic p if p does not divide the group order then molien s formula gives the hilbert series of the invariant ring in this paper we find a replacement of molien s formula which works in the case that g is divisible by p but not by p 2 we also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of v into indecomposables our methods can be applied to determine the depth of the invariant ring without computing any invariants this leads to a proof of a conjecture of the second author on certain invariants of gl 2 p

a discretization of p adic quantum mechanics we show that some compact subgroups n m of thep adic heisenberg group act irreducibly on corresponding finite dimensional spaces of test functions sm n under certain conditions a compact group am n of linear canonical transformations isomorphic tosl 2 zp can be represented unitarily onsm n as a group of automorphisms of n m the restriction tosm n can be considered as a discretization because an invariant subgroup in m ofam n is represented trivially it is possible to take a limit whereim n becomes an arbitrarily small neighborhood of the identity while the dimension ofsm n becomes arbitrarily large this is a possible definition of the continuum limit that we relate to other projective limits appearing naturally in the present context

mapping toric varieties into low dimensional spaces a smooth d dimensional projective variety x can always be embedded into 2d 1 dimensional space in contrast a singular variety may require an arbitrary large ambient space if we relax our requirement and ask only that the map is injective then any d dimensional projective variety can be mapped injectively to 2d 1 dimensional projective space a natural question then arises what is the minimal m such that a projective variety can be mapped injectively to m dimensional projective space in this paper we investigate this question for normal toric varieties with our most complete results being for segre veronese varieties

some complete intersection symplectic quotients in positive characteristic invariants of a vector and a covector abstract given a linear action of a group g on a k vector space v we consider the invariant ring k v v g where v is the dual space we are particularly interested in the case where v f q n and g is the group u n of all upper unipotent matrices or the group b n of all upper triangular matrices in gl n f q in fact we determine f q v v g for g u n and g b n the result is a complete intersection for all values of n and q we present explicit lists of generating invariants and their relations this makes an addition to the rather short list of doubly parametrized series of group actions whose invariant rings are known to have a uniform description

multiloop calculations in p adic string theory and bruhat tits trees we treat the openp adic string world sheet as a coset spacef t wheret is the bruhat tits tree for thep adic linear groupgl 2 p and pgl 2 p is some schottky group the boundary of this world sheet corresponds to ap adic mumford curve of finite genus the string dynamics is governed by the local gaussian action on the coset spacef the tachyon amplitudes expressed in terms ofp adic functions are proposed for the mumford curve of arbitrary genus we compare them with the corresponding usual archimedean amplitudes the sum over moduli space of the algebraic curves is conjectured to be expressed in the arithmetic surface terms we also give the necessary mathematical background including the mumford approach top adic algebraic curves the connection of the problem of closedp adic strings with the considered topics is discussed

on the cohen macaulay property of multiplicative invariants we investigate the cohen macaulay property for rings of invariants under multiplicative actions of a finite group g by definition these are g actions on laurent polynomial algebras k x 1 1 x 1 n that stabilize the multiplicative group consisting of all monomials in the variables x i for the most part we concentrate on the case where the base ring k is z our main result states that if g acts non trivially and the invariant ring z x 1 1 x 1 n g is cohen macaulay then the abelianized isotropy groups g ab m of all monomials m are generated by the bireflections in p m and at least one g ab m is non trivial as an application we prove the multiplicative version of kemper s 3 copies conjecture

hierarchical structures and asymmetric stochastic processes on p adics and adeles the space of states of some phenomena in physics and other sciences displays a hierarchical structure when that is the case it is natural to label the states by a p adic number field both the classification of the states and their relationships are then based on a notion of distance with ultrametric properties the dynamics of the phenomena that is the transition between different states is also a function of the p adic distance dp however because the distance is a symmetric function probabilistic processes which depend only on dp have a uniform invariant probability measure that is all states are equally probable at large times this being a severe limitation for cases of physical interest processes with asymmetric transition functions have been studied in addition to the dependence on the ultrametric distance the asymmetric transition functions are allowed to depend also on the probability of the target state leading to any desired invariant probability measure when each state of a phy

flips from 4 folds with isolated complete intersection singularities we investigate a flipping contraction g x y from a 4 fold x with at worst isolated complete intersection singularities if y has an anti bicanonical divisor bi elephant with only rational singularities then g carries an inductive structure involving a chain of blow ups which we call la torre pendente and in particular the flip exists this naturally contains reid s pagoda as an anticanonical divisor elephant and its proper transforms

two dimensional vector invariant rings of abelian p groups abstract for any faithful representation v of a non trivial p group over a field of characteristic p 0 it is known that the ring of vector invariants of m copies of v is not cohen macaulay if m 3 however much less is known about the case m 2 in this paper we show that if m 2 and the group is an abelian p group then the ring of invariants of 2v is a complete intersection in some cases and is not cohen macaulay in most cases as a corollary we obtain that if the field is f p and the ring of invariants of the representation v is a polynomial ring then the ring of invariants of 2v is either a complete intersection or not cohen macaulay

non cohen macaulay invariant rings of infinite groups abstract we give explicit examples of invariant rings that are not cohen macaulay for all classical groups sl n k gl n k sp 2 n k so n k and o n k where k is an algebraically closed field of positive characteristic we prove that every non trivial unipotent group over k has representations such that the invariant ring is not cohen macaulay

scattering on p adic graphs abstract the scattering problem on the bruhat tits tree and its quotient spaces realizing the p adic riemann surfaces is studied the spectral decomposition of the corresponding laplace beltrami operator is constructed the stationary s matrix is obtained and the lax phillips scattering theory for the problem is developed in a closed form the eisenstein series technique is applied to 1 loop case the analytical structure of the scattering matrix for joint spherical functions is described

on the depth of invariant rings of infinite groups abstract let k be an algebraically closed field for a finitely generated graded commutative k algebra r let cmdef r dim r depth r denote the cohen macaulay defect of r let g be a linear algebraic group over k that is reductive but not linearly reductive we show that there exists a faithful rational representation v of g which we will give explicitly such that cmdef k v k g k 2 for all k n

arrangements defined by unitary reflection groups let v be a complex vector space of dimension i an arrangement in v is a finite set d of hyperplanes all containing the origin let l l d be the set of intersections of elements of r partially order l by reverse inclusion so that l has v as its minimal element and d as its set of atoms the poset l is a finite geometric lattice with rank function r x dim v x xe l without loss of generality we assume that n h h 0 is the maximal element of l and thus l has rank i the characteristic polynomial z l t of l is defined by

ising model on the generalized bruhat tits tree the partition function and the correlation functions of the ising model on the generalized bruhat tits tree are calculated we computed also the averages of these correlation functions when the corresponding vertices are attached to the boundary of the generalized bruhat tits tree

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

an algebraic model for chains on ωbg p we provide an interpretation of the homology of the loop space on the p completion of the classifying space of a finite group in terms of representation theory and demonstrate how to compute it we then give the following reformulation if f is an idempotent in kg such that f kg is the projective cover of the trivial module k and e 1 f then we exhibit isomorphisms for n 2 h n bg p k tor e kg e n 1 kg e e kg h n bg p k ext n 1 e kg e e kg e kg further algebraic structure is examined such as products and coproducts restriction and steenrod operations

on cohen macaulay rings of invariants abstract we investigate the transfer of the cohen macaulay property from a commutative ring to a subring of invariants under the action of a finite group our point of view is ring theoretic and not a priori tailored to a particular type of group action as an illustration we discuss the special case of multiplicative actions that is actions on group algebras k z n via an action on z n

separating invariants for 2 2 matrices abstract a minimal separating set is found for the algebra of matrix invariants of several 2 2 matrices over an infinite field of arbitrary characteristic

separating invariants and local cohomology the study of separating invariants is a recent trend in invariant theory for a finite group acting linearly on a vector space a separating set is a set of invariants whose elements separate the orbits of g in some ways separating sets often exhibit better behavior than generating sets for the ring of invariants we investigate the least possible cardinality of a separating set for a given g action our main result is a lower bound that generalizes the classical result of serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections we find these bounds to be sharp in a wide range of examples

the mumford form and the polyakov measure in string theory an explicit formula is derived for the mumford form on the moduli space of algebraic curves according to the belavin knizhnik theorem this gives a formula for the polyakov bosonic string measure

noncommutative complete intersections abstract several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded k algebra these notions are justified by examples from noncommutative invariant theory

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

separating invariants of finite groups this thesis studies separating invariants of finite algebraic groups acting on affine varieties through automorphisms we investigate the question what restrictions does the existence of a separating set of small size or a separating algebra with interesting structural properties impose on the group action theorems of serre dufresne kac watanabe and gordeev and dufresne and jeffries on invariant rings and separating algebras of linear representations are extended to this more general situation

rings of invariants of certainp groups over the fieldfp abstract we use nakajima s j algebra85 1983 253 286 characterization ofp groups with polynomial rings of invariants over the fieldfpto prove that maximal proper subgroups of such groups have rings of invariants that are hypersurfaces we give an explicit construction for the ring of invariants of such a group as a module over the ring of invariants of the total group

depth and detection in modular invariant theory let g be a finite group acting linearly on a vector space v over a field of characteristic p dividing the group order and let r denote s v we study the r g modules h i g r for i 0 with r g itself as a special case there are lower bounds for depth of h i g r and for depth r g we show that a certain sufficient condition for their attainment due to fleischmann kemper and shank may be modified to give a condition which is both necessary and sufficient we apply our main result to classify the representations of the klein four group for which depth r g attains its lower bound we also use our new condition to show that the if g p q with p a p group and q an abelian p group then the depth of r g attains its lower bound if and only if the depth of r p does so

quotients et extensions de groupes de reflexions complexes we give a geometric description of a certain class of epimorphisms between complex reflection groups we classify these epimorphisms which can be interpreted as morphisms between the diagrams symbolizing standard presentations by generators and relations for complex reflection groups and their braid groups

macdonald polynomials from sklyanin algebras a conceptual basis for the p adics quantum group connection we establish a previously conjectured connection betweenp adics and quantum groups we find in sklyanin s two parameter elliptic quantum algebra and its generalizations the conceptual basis for the macdonald polynomials which interpolate between the zonal spherical functions of related real andp adic symmetric spaces the elliptic quantum algebras underlie thez n baxter models we show that in then limit the jost function for the scattering offirst level excitations in the 1 1 dimensional field theory model associated to thez n baxter model coincides with the harish chandra likec function constructed from the macdonald polynomials associated to the root systema 1 the partition function of thez 2 baxter model itself is also expressed in terms of this macdonald harish chandrac function albeit in a less simple way we relate the two parametersq andt of the macdonald polynomials to the anisotropy and modular parameters of the baxter model in particular thep adic regimes in the macdonald polynomials correspond to a discrete sequence of xxz models we also discuss the possibility of q deforming euler products

separating invariants and finite reflection groups abstract a separating algebra is roughly speaking a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants in this paper we introduce a geometric notion of separating algebra this allows us to prove that only groups generated by reflections may have polynomial separating algebras and only groups generated by bireflections may have complete intersection separating algebras

multilevel simultaneous equation model a novel specification and estimation approach abstract conventional simultaneous equation models assume that the error terms are serially independent in some situations data may present hierarchical or grouped structure and this assumption may be invalid a new multivariate model referred as to multilevel simultaneous equation model msem is developed under this motivation the maximum likelihood estimation of the parameters of an msem is considered a matrix valued distribution namely the matrix normal distribution is introduced to incorporate an among row and an among column covariance matrix structure in the specification of the model in the absence of an analytical solution of the system of likelihood equations a general purpose optimization solver is employed to obtain the maximum likelihood estimators in a first approach to the solution of the problem the adequacy of the matrix normal distribution is evaluated empirically in the case in which the double covariance structure is known using simulated data under the model assumptions the performance of the maximum likelihood estimator mle is assessed with regard to other conventional alternatives such as two stage least squares estimator 2sls

on the branch locus of quotients by finite groups and the depth of the algebra of invariants abstract let a bg where b is a noetherian algebra over a field k of characteristic p 0 and g is a finite group such that p divides g we give estimates for the depth of a in terms of the codimension of the branch locus of the extension b a

constructing modular separating invariants abstract we consider a finite dimensional modular representation v of a cyclic group of prime order p we show that two points in v that are in different orbits can be separated by a homogeneous invariant polynomial that has degree one or p and that involves variables from at most two summands in the dual representation simultaneously we describe an explicit construction for a separating set consisting of polynomials with these properties

invariants and separating morphisms for algebraic group actions the first part of this paper is a refinement of winkelmann s work on invariant rings and quotients of algebraic group actions on affine varieties where we take a more geometric point of view we show that the algebraic quotient x g given by the possibly not finitely generated ring of invariants is almost an algebraic variety and that the quotient morphism pi x rightarrow x g has a number of nice properties one of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective these general results are then refined for actions of the additive group mathbb g a where we can say much more we get a rather explicit description of the so called plinth variety and of the separating variety which measures how much orbits are separated by invariants the most complete results are obtained for representations we also give a complete and detailed analysis of roberts famous example of a an action of mathbb g a on 7 dimensional affine space with a non finitely generated ring of invariants

loci in quotients by finite groups pointwise stabilizers and the buchsbaum property let k v g be the invariant ring of a nite linear group g gl v and let gu be the pointwise stabilizer of a subspace u v we prove that the following numbers associated to the invariant ring do not increase if one passes from k v g to k v gu the minimal number of homogeneous generators the maximal degree of the generators the number of syzygies and other betti numbers the complete intersection defect the dierence between depth and dimension and the type from this theorems of steinberg serre nakajima kac and watanabe and the author follow which say that if k v g is a polynomial ring a hypersurface a complete intersection or cohen macaulay then the same is true for k v gu furthermore k v gu inherits the gorenstein property from k v g we give an algorithm which transforms generators of k v g into generators of k v g u letp be one of the properties mentioned above we consider the locus ofp in v g spec k v g and prove that for x2 spec k v with image x 0 in v g the local ring k v g0 has the property p if and only if p holds for the invariant ring k v g x of the point stabilizer using this we prove that the non cohen macaulay locus in v g is either empty or it has dimension at least one and codimension at least 3 from this we deduce that k v g is buchsbaum if and only if it is cohen macaulay this proves a conjecture of campbell et al

non archimedean strings and bruhat tits trees it is shown that the bruhat tits tree for thep adic linear groupgl 2 is a natural non archimedean analog of the open string world sheet the boundary of the tree can be identified with the field ofp adic numbers we construct a lattice quantum field theory on the bruhat tits tree with a simple local lagrangian and show that it leads to the freund olson amplitudes for emission processes of the particle states from the boundary

p adic ads cft we construct a p adic analog to ads cft where an unramified extension of the p adic numbers replaces euclidean space as the boundary and a version of the bruhat tits tree replaces the bulk correlation functions are computed in the simple case of a single massive scalar in the bulk with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions we give some brief discussion of the geometry of p adic chordal distance and of wilson loops our presentation includes an introduction to p adic numbers