decoherence by lindblad motion the predictive power of lindblad equations for the dynamics of open systems is discussed in a model with only one lindblad operator the asymptotic state density operator for t determines the lindblad operator up to an isometry hence the asymptotic state reached by lindblad motion appears as an input and has to be obtained from independent physical considerations when setting up the equations of motion to illustrate this fact we assume an asymptotic gibbs state and a hamiltonian of course and discuss the temporal behaviour of the statistical entropy numerical model calculations are presented which show a non monotonic behaviour of the statistical entropy during the approach to equilibrium as a second point in our discussion of the types of predictions derived from lindblad equations we show that a certain structure of the lindblad operators leads to decoherence into superselection sectors the latter are determined by a spectral decomposition of the hamiltonian of the open system considered an explicit construction of such decohering systems is given

analogue of the bruhat chevalley order for reductive monoids the purpose of this paper is to describe the adherence order of w x b b orbits on a reductive algebraic monoid by the results of 10 there is already a perfect analogue of the bruhat decomposition for reductive w x monoids indeed by 10 corollary 5 8 if m is reductive with unit group g and maximal torus t g with borel subgroup b t then the set of two sided b orbits b r mrb is canonically identified with a certain finite inverse monoid r in fact r s n t rt the orbit monoid of the zariski g closure of n t in m by definition for s t g r we define the adherg ence order by s f t if bs b bt b where bt b denotes the zariski closure in m of bt b

the total coordinate ring of a wonderful variety abstract we study the cone of effective divisors and the total coordinate ring of wonderful varieties with applications to their automorphism group we show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings

monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

tian s alpha _ m k hat k invariants on group compactifications in this paper we compute tian s alpha m k k times k invariant on a polarized g group compactification where k denotes a maximal compact subgroup of a connected complex reductive group g we prove that tian s conjecture see conjecture 1 1 below is true for alpha m k k times k invariant on such manifolds when k 1 but it fails in general by producing counter examples when k ge 2

unitary scaling decomposition and dissipative behaviour in finite dimensional unital lindblad dynamics abstract we investigate a decomposition of a unital lindblad dynamical map of an open quantum system into two distinct types of mapping on the hilbert schmidt space of quantum states one component of the decomposed map corresponds to reversible behaviours while the other to irreversible characteristics for a finite dimensional system we employ real vectors or bloch representations and express a dynamical map on the state space as a real matrix acting on the representation it is found that rotation and scaling transformations on the real vector space obtained from the real polar decomposition form building blocks for the dynamical map consequently the change of the linear entropy or purity which indicates dissipative behaviours depends only on the scaling part of the dynamical matrix the rate of change of the entropy depends on the structure of the scaling part of the dynamical matrix such as eigensubspace partitioning and its relationship with the initial state in particular the linear entropy is expressed as a weighted sum of the exponential decay functions in each scaling component where the weight is equal to x k 2 of the initial state in the subspace the dissipative behaviours and the partition of eigensubspaces in the decomposition are discussed and illustrated for qubit systems

control of quantum states in decoherence free subspaces the problem of quantum state control in open quantum systems based on the lyapunov method is studied the control laws are designed to drive the system to a pure target state in decoherence free subspace dfs in the interaction picture where the average of an observable operator p is chosen as a lyapunov function by analyzing the positive limit set via barbalat s lemma we propose a sufficient condition on the observable operator such that the positive limit set only contains the target state under this condition it is proved that the control laws can drive the open system to any pure target state in dfs in the interaction picture also we provide a method to construct the observable operator via schmidt orthogonalization numerical simulations on a three level system validate the effectiveness of the proposed control strategy

uber bewertungen welche unter einer reduktiven gruppe invariant sind letg be a reductive group defined over an algebraically closed fieldk and letx be ag variety in this paper we studyg invariant valuationsv of the fieldk of rational functions onx these objects are fundamental for the theory of equivariant completions ofx letb be a borel subgroup andu the unipotent radical ofb it is proved thatv is uniquely determined by its restriction toku then we study the set of invariant valuations having some fixed restrictionv0 tokb ifv0 is geometric i e induced by a prime divisor then this set is a polyhedron in some vector space in characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupwx thus the classification of invariant valuations is almost reduced to the classification of valuations ofkb

unstable asymptotics for nonstationary queues we relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes we apply these results to develop approximations for queues that are unstable in a nonstationary manner we obtain unstable nonstationary analogs of the queueing relation l w and associated central limit theorem versions for modeling and to obtain the first limits we can construct nonstationary point processes as random time transformations of familiar point processes such as renewal processes and stationary point processes we deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation

conjugacy classes of renner monoids abstract in this paper we describe conjugacy classes of a renner monoid r with unit group w the weyl group we show that every element in r is conjugate to an element ue where u w and e is an idempotent in a cross section lattice denote by w e and w e the centralizer and stabilizer of e in w respectively let w e act by conjugation on the set of left cosets of w e in w we find that ue and ve u v w are conjugate if and only if u w e and v w e are in the same orbit as consequences there is a one to one correspondence between the conjugacy classes of r and the orbits of this action we then obtain a formula for calculating the number of conjugacy classes of r and describe in detail the conjugacy classes of the renner monoid of some j irreducible monoids we then generalize munn conjugacy on a rook monoid to any renner monoid and show that munn conjugacy coincides with semigroup conjugacy action conjugacy and character conjugacy we also show that the number of inequivalent irreducible representations of r over an algebraically closed field of characteristic zero equals the number of munn conjugacy classes in r

complex representations of matrix semigroups let m be a finite monoid of lie type these are the finite analogues of linear algebraic monoids with group of units g the multiplicative semigroup 4 f where f is a finite field is a particular example using harishchandra s theory of cuspidal representations of finite groups of lie type we show that every complex representation of m is completely reducible using this we characterize the representations of g extending to irreducible representations of m as being those induced from the irreducible representations of certain parabolic subgroups of g we go on to show that if f is any field and s any multiplicative subsemigroup of 4 f then the semigroup algebra of s over any field of characteristic zero has nilpotent jacobson radical if s 4 f then this algebra is jacobson semisimple finally we show that the semigroup algebra of 4 f over a field of characteristic zero is regular if and only if ch f p 0 and f is algebraic over its prime field

k energy on polarized compactifications of lie groups abstract in this paper we study mabuchi s k energy on a compactification m of a reductive lie group g which is a complexification of its maximal compact subgroup k we give a criterion for the properness of k energy on the space of k k invariant kahler potentials in particular it turns to give an alternative proof of delcroix s theorem for the existence of kahler einstein metrics in case of fano manifolds m we also study the existence of minimizers of k energy for general kahler classes of m

analysis of quantum semigroups with gks lindblad generators i simple generators semigroups describing the time evolution of open quantum systems in finite dimensional spaces have generators of a special form known as lindblad generators the simple generators characterized by only one operator are analyzed the complete set of all the stationary states is presented in detail including a formula to calculate a stationary state from the generating operator also the opposite task can be fulfilled to construct an evolution leading to a prescribed stationary state

modeling subdiffusion using reaction diffusion systems in many types of media particularly within living cells or within their membranes diffusing species do not follow fick s laws but instead show transient subdiffusive behavior formulating spatiotemporal models that take this behavior into account is a delicate matter as one is faced with the choice of resorting either to fractional calculus or to microscopic descriptions in this article we provide an alternative designed to be easier to handle analytically and numerically than the existing approaches specifically starting from the continuous time random walk model we construct linear reaction diffusion systems that can be used as components within such a model and which capture the defining properties of subdiffusion we show how to impose physically relevant parameters and prove stability and mass conservation while applications to cellular biology are our main motivation our approach is abstract and should thus be applicable to any situation where anomalous subdiffusion is observed

the lattice of j classes of j σ irreducible monoids ii abstract a reductive monoid is an algebraic monoid with a reductive unit group generally we have the question of finding the orbits of the unit group of a reductive monoid acting on both sides of the monoid putcha and renner give a recipe to determine the orbits for j irreducible monoids according to the dynkin diagrams we obtain a similar recipe for the question to j irreducible monoids not j irreducible of type d 2 n however there is no similar answer for types a n n 4 and e 2 6 the fixed points of any j irreducible monoid under is a finite reductive monoid we obtain that any such finite reductive monoid is j irreducible then we find the orbits of these monoids under the two sided action of their unit groups

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

decompositions of hilbert spaces stability analysis and convergence probabilities for discrete time quantum dynamical semigroups we investigate convergence properties of discrete time semigroup quantum dynamics including asymptotic stability probability and speed of convergence to pure states and subspaces these properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing our results include two hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence as well as a formula to obtain the limit probability distribution for a set of orthogonal invariant subspaces

the physics of the m t g symbol queue we establish some general structural results and derive some simple formulas describing the time dependent performance of the mt g queue with a nonhomogeneous poisson arrival process we know that for appropriate initial conditions the number of busy servers at time t has a poisson distribution for each t our results show how the time dependent mean function m depends on the time dependent arrival rate function and the service time distribution for example when is quadratic the mean m t coincides with the pointwise stationary approximation t e s where s is a service time except for a time lag and a space shift it is significant that the well known insensitivity property of the stationary m g model does not hold for the nonstationary mt g model the time dependent mean function m depends on the service time distribution beyond its mean the service time stationary excess distribution plays an important role when is decreasing before time t m t is increasing in the service time

the h polynomial of an irreducible representation abstract let g be a simple algebraic group associated with the finite dimensional rational representation g end v of g there is the monoid m k g end v and the projective g g embedding p m 0 k one can identify the cases where p is rationally smooth and in such cases it is desirable to calculate the h polynomial h of p in this paper we consider the situation where is irreducible we then determine h explicitly in terms of combinatorial invariants of indeed there is a canonical cellular decomposition for p these cells are defined in terms of idempotents b b orbits and other natural quantities obtained from m furthermore h is obtained by recording the dimension of each of these cells in terms of the descent system of m as a special case we reacquire the well known formula for the poincare polynomial of a wonderful embedding of a simple algebraic group of adjoint type

the h polynomial of a semisimple monoid abstract a semisimple monoid m is called quasismooth if m 0 has sufficiently mild singularities we define a cellular decomposition of such monoids using the method of one parameter subgroups these cells turn out to be almost affine spaces but they can also be described in terms of the idempotents and b b orbits of m this leads to a number of combinatorial results about the inverse monoid of b b orbits of m in particular we obtain fundamental information about the h polynomial of m

asymptotic expansions of singularly perturbed systems involving rapidly fluctuating markov chains a class of singularly perturbed time varying systems with a small parameter varepsilon 0 is considered in this paper the importance of the study stems from the fact that many problems arise in various applications involve a rapidly fluctuating markov chain to investigate the limit behavior of such systems it is necessary to consider the corresponding singular perturbation problems existing results in singular perturbation of ordinary differential equations cannot be applied since the coefficient matrix of the equation is a generator of a finite state markov chain and as a result it is singular asymptotic properties of the aforementioned systems are developed via matched asymptotic expansion in this paper thanks to the specific structure and the properties of markovian generators it is established that the solution of the system can be approximated as close as possible by a series expansion in terms of the small parameter varepsilon 0

asymptotic analysis of a state dependent m g 1 queueing system we present new asymptotic methods for the analysis of queueing systems these methods are applied to a state dependent m g 1 queue we formulate problems for and compute approximations to i the stationary density of the unfinished work ii the mean length of time until the end of a busy period iii the mean length of a busy period and iv the mean time until the unfinished work reaches or exceeds a specified capacity the methods are applied to the full kolmogorov equations scaled so that the arrival rate is rapid and the mean service is small thus we do not truncate equations as in diffusion approximations for state independent m g 1 queues our results are shown to agree with the known exact solutions we include comparisons both analytic and numerical between our results and those obtained from diffusion approximations

idempotent lattices renner monoids and cross section lattices of the special orthogonal algebraic monoids abstract let mson n is even be the special orthogonal algebraic monoid t a maximal torus of the unit group and t the zariski closure of t in the whole matrix monoid m n in this paper we explicitly determine the idempotent lattice e t the renner monoid r and the cross section lattice of mson in terms of the weyl group and the concept of admissible sets see definition 3 1 it turns out that there is a one to one relationship between e t and the admissible subsets and that r is a submonoid of r n the renner monoid m n also is a sublattice of n the cross section lattice of m n

the canonical compactification of a finite group of lie type let g be a finite group of lie type we construct a finite monoid m having g as the group of units m has properties analogous to the canonical compactification of a reductive group the complex representation theory of m yields harish chandra s philosophy of cuspidal representations of g the main purpose of this paper is to determine the irreducible modular representations of m we then show that all the irreducible modular representations of g come via the 1942 work of clifford from the one dimensional representations of the maximal subgroups of m this yields a semigroup approach to the modular representation theory of g via the full rank factorizations of the sandwich matrices of m we then determine the irreducible modular representations of any finite monoid of lie type

on categories of equivariant d modules abstract let x be a variety with an action by an algebraic group g in this paper we discuss various properties of g equivariant d modules on x such as the decompositions of their global sections as representations of g when g is reductive and descriptions of the categories that they form when g acts on x with finitely many orbits the category of equivariant d modules is equivalent to the category of finite dimensional representations of a finite quiver with relations we describe explicitly these categories for irreducible g modules x that are spherical varieties and show that in such cases the quivers are almost always representation finite i e with finitely many indecomposable representations

classification of semisimple algebraic monoids let x be a semisimple algebraic monoid with unit group g associated with e is its polyhedral root system x c where x x t is the character group of the maximal torus t c g c x t is the set of roots and c x t is the character monoid of t c e zariski closure the correspondence e x c is a complete and discriminating invariant of the semisimple monoid e extending the well known classification of semisimple groups in establishing this result monoids with preassigned root data are first constructed from linear representations of g that done we then show that any other semisimple monoid must be isomorphic to one of those constructed to do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory thls ultimately yields the desired conclusions consider the classification problem for semisimple algebraic monoids over the algebraically closed field k what sort of problem is this first the definition a semisimple algebraic monoid is an irreducible affine algebraic variety e defined over k together with an associative morphlsm m e x e e and a two sided unit 1 e e for m we assume further that e has a 0 the unit group g which is always linear algebraic and dense in e is reductive e g gl k dim zg 1and e is a normal variety the problem then presents us with two familiar objects let t c g be a maximal torus then z t c e zariski closure is an affine torus embedding and g c e is a reductive algebraic group torus embeddings have been introduced by demazure in 6 in his study of cremona groups and are classified numerically using rational polyhedral cones 14 on the other hand reductive groups have been studied at least in principle since the nineteenth century their classification in the modern sense being achieved largely by chevalley 4 that numerical classification uses the now familiar root systems l l chapter 31 of killing that were introduced by him 15 in his penetrating formulation of the classification e cartan s of semisimple lie algebras thus in the classification of semisimple monoids we are compelled to consider the root system x x t t of g and the polyhedral cone c x z c x t of t c z the two objects are canonically related by the weyl group action on x which leaves c stable received by the editors november 16 1984 1980 mathematics subject classification primary 14m99 secondary 20m99

infinite dimensional reductive monoids associated to highest weight representations of kac moody groups abstract starting with a highest weight representation of a kac moody group over the complex numbers we construct a monoid whose unit group is the image of the kac moody group under the representation multiplied by the nonzero complex numbers we show that this monoid has similar properties to those of a j irreducible reductive linear algebraic monoid in particular the monoid is unit regular and has a bruhat decomposition and the idempotent lattice of the generalized renner monoid of the bruhat decomposition is isomorphic to the face lattice of the convex hull of the weyl group orbit of the highest weight

time dependent queues a single server queue is considered having exponentially distributed inter arrival and service times with slowly changing time dependent rates lambda varepsilon t and mu varepsilon t the parameter varepsilon is the ratio of an average inter arrival time to the time over which the rates change appreciably so it is small therefore an asymptotic solution valid for varepsilon small is constructed for the time dependent queue length probability distribution it consists of five typical parts corresponding to five typical time periods they are the initial period the period of light traffic when lambda mu 1 the saturation transition period when lambda mu increases through unity the oversaturation period when lambda mu starts out greater than unity and then decreases below unity and the transition period at the end of oversaturation when the queue returns to the light traffic condition by combining the solutions for these five intervals the solution fo

two time scale markov chains and applications to quasi birth death queues aiming at reduction of complexity this work is concerned with two time scale markov chains and applications to quasi birth death queues asymptotic expansions of probability vectors are constructed and justified lumping all states of the markov chain in each subspace into a single state an aggregated process is shown to converge to a continuous time markov chain whose generator is an average with respect to the stationary measures then a suitably scaled sequence is shown to converge to a switching diffusion process extensions of the results are presented together with examples of quasi birth death queues

split subvarieties of group embeddings let g be a connected reductive group and x an equivariant com pactifiction of g in x we study generalised and opposite generalised schu bert varieties their intersections called generalised richardson varieties and projected generalised richardson varieties any complete g embedding has a canonical frobenius splitting and we prove that the compatibly split subvari eties are the generalised projected richardson varieties extending a result of knutson lam and speyer to the situation

irreducible representations of semisimple algebraic groups and supersolvable lattices abstract let be the cross section lattice of an irreducible representation of a semisimple algebraic group certain combinatorial properties of are studied supersolvable s are determined in terms of dynkin diagrams the characteristic polynomial of a supersolvable is computed

kahler geometry of horosymmetric varieties and application to mabuchi s k energy functional we introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties we develop kahler geometry on these varieties with applications to canonical metrics in mind as a generalization of the guillemin abreu donaldson geometry of toric varieties namely we associate convex functions with hermitian metrics on line bundles and express the curvature form in terms of this function as well as the corresponding monge ampere volume form and scalar curvature we then provide an expression for the mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by li zhou and zhu on group compactifications

asymptotic analysis of the time dependent m m 1 queue using operator analytic techniques we develop a nonstationary markovian queueing theory starting with the mt mt 1 queue we employ an asymptotic approach quite different from the usual large time analysis instead we uniformly accelerate the queue length process that is we divide the arrival and service rate by a common parameter e then for a fixed time interval we consider the asymptotics for the distribution mean and variance of the queue length process as e goes to zero the effects of e can be quite different for the given time interval this gives us a dynamic asymptotic behavior for the queue length process we can formulate a time dependent traffic intensity parameter that determines when the process is asymptotically stable and when it is asymptotically unstable

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

complete moduli in the presence of semiabelian group action i prove the existence and describe the structure of moduli space of pairs p consisting of a projective variety p with semiabelian group action and an ample cartier divisor on it satisfying a few simple conditions every connected component of this moduli space is proper a component containing a projective toric variety is described by a configuration of several polytopes the main one of which is the secondary polytope on the other hand the component containing a principally polarized abelian variety provides a moduli compactification of a g the main irreducible component of this compactification is described by an infinite periodic analog of the secondary polytope and coincides with the toroidal compactification of a g for the second voronoi decomposition

asymptotic expansions of solutions of integro differential equations for transition densities of singularly perturbed switching diffusions rapid switchings this work develops asymptotic expansions for solutions of integro differential equations arising from transition densities of singularly perturbed switching diffusion processes the underlying processes have two components namely the switching component and the diffusion component processes with rapid switchings are examined here under suitable conditions it is proved that the asymptotic expansion is uniformly valid in the domain of interest

analog of the bruhat decomposition for algebraic monoids ii the length function and the trichotomy 0 introduction the purpose of this paper is to continue the study of b b orbits on a reductive monoidm this was first taken up systematically by the author in 9 where several basic properties analogous to the case of groups were established the b b orbits were identified with a finite monoid r defined in terms of the normalizer of a maximal torus and a direct analysis of these orbits led to an extension of tits axiom br brb b rb to the case of reductive monoids however there is more to this axiom when dealing with monoids obviously

the i class structure of connected algebraic monoids abstract let s be a connected algebraic monoid and let u s denote the finite lattice of all regular i classes of s the main theorem of this paper is that u s is relatively complemented if and only if s is a semilattice of archimedean semigroups if further s has a zero then u s is relatively complemented if and only if the group of units g of s is solvable another characterization for the solvability of g is that the product of any two nilpotent elements of s is again nilpotent the width and depth of any j u s are defined also the width depth and height of s are defined these structure numbers are studied in some detail and the solvability of g is characterized in terms of them

analysis of quantum semigroups with gks lindblad generators ii general semigroups describing the time evolution of open quantum systems in finite dimensional spaces have generators of a special form known as lindblad generators these generators and the corresponding processes of time evolution are analyzed and characterized as decay dissipation and dephasing in relation to these processes the hilbert space of the system is equipped with a special structure a decomposition into a sum of mutually orthogonal subspaces the complete set of all the stationary states and the asymptotic behavior of the evolutions are presented in detail some unusual special facts about invariant operators and symmetries are studied and examples are demonstrated perturbation theory for the structure and for the stationary states is discussed and performed in case studies

heavy traffic analysis of the sojourn time in tandem queues with overtaking a tandem jackson network with a processor sharing ps node followed by either another ps node or a first in first out fifo node is considered it is assumed that both nodes are in heavy traffic three term asymptotic approximations to the distribution of the total sojourn response time through the two nodes are computed the leading term in the asymptotic approximation corresponds to the heavy traffic limit in which the effects of overtaking disappear the correction terms in the approximations give a quantitative measure of the erects of overtaking which occurs due to the presence of the ps node s

types of reductive monoids abstract let m be a reductive monoid with a reductive unit group g clearly there is a natural g g action on m the orbits are the j classes in the sense of semigroup theory and form a finite lattice the general problem of finding the lattice remains open in this paper we study a new class of reductive monoids constructed by multilined closure we obtain a general theorem to determine the lattices of these monoids we find that the j irreducible monoids of suzuki type and ree type belong to this new class using the general theorem we then list all the lattices and type maps of the j irreducible monoids of suzuki type and ree type

shellability in reductive monoids the purpose of this paper is to extend to monoids the work of bjorner wachs and proctor on the shellability of the bruhat chevalley order on weyl groups let m be a reductive monoid with unit group g borel subgroup b and weyl group w we study the partially ordered set of b x b orbits with respect to zariski closure inclusion within a g x g orbit of m this is the same as studying a w x w orbit in the renner monoid r such an orbit is the retract of a universal orbit which is shown to be lexicograhically shellable in the sense of bjorner and wachs

stable reductive varieties ii projective case abstract we construct a moduli space of stable projective pairs with a nontrivial action of a connected reductive group these stable reductive pairs are higher dimensional analogs of stable n pointed curves and generalize to the non commutative case a functorial compactification of moduli of abelian varieties and of toric pairs we prove that confirming a prediction of log minimal program stable reductive pairs have semi log canonical singularities

reductive monoids are von neumann regular let k be an algebraically closed field an a e algebraic monoid e is an affme algebraic variety defined over k together with an associative morphism m e x e e and a two sided unit 1 e e for m in categorical terminology an afftne algebraic monoid is a representable functor from the category of afftne varieties to the category of monoids algebraic group theory is concerned mainly with reductive groups and their representations whereas the theory of monoids is concerned largely with von neumann regular monoids the main result of this paper provides a fundamental link between these two objects

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

weylgruppe und momentabbildung let g be a connected reductive group defined over an algebraically closed field of characteristic zero we assign to any g variety x a finite cristallographic reflection group w x by means of the moment map on the cotangent bundle this generalizes the little weyl group of a symmetric space the weyl group w x is related to the equivariant compactification theory of x we determine the closure of the image of the moment map and the generic isotropy group of the action of g on the cotangent bundle as a byproduct we determine the ideal of elements of mathfrak u g which act trivially on x as a differential operator

orders of the renner monoids abstract in this paper we give a general formula for a renner monoid of a reductive monoid with zero it is based on putcha s type map we apply this formula to find the orders of the renner monoids for j irreducible monoids k g where g is a simple algebraic group over an algebraically closed field k and g gl v is an irreducible representation associated with an arbitrary fundamental dominant weight