applications of the brauer complex card shuffling permutation statistics and dynamical systems by algebraic group theory there is a map from the semisimple conjugacy classes of a finite group of lie type to the conjugacy classes of the weyl group picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the weyl group using the brauer complex it is proved that this measure agrees with a second measure on conjugacy classes of the weyl group induced by a construction of cellini using the affine weyl group formulas for cellini s measure in type a are found this leads to new models of card shuffling and has interesting combinatorial and number theoretic consequences an analysis of type c gives another solution to a problem of rogers in dynamical systems the enumeration of unimodal permutations by cycle structure the proof uses the factorization theory of palindromic polynomials over finite fields contact is made with symmetric function theory

interpolation norms between row and column spaces and the norm problem for elementary operators abstract for the class of transformers acting as x a t x b t d t on the space of bounded hilbert space operators we give formulae for its norm on the hilbert schmidt class 1 x a t x b t d t b c 2 h lim n 2 n tr k 1 n a t n 1 k a s n 1 k tr k 1 n b s k b t k k 1 n d s k d t k 2 n whenever a t p b t p d t for some p 0 we also estimate from below its norm on the other schatten classes this answers a question of characterizing 1 2 interpolation norm between column and row space norm for operator valued functions with the discrete case providing the solution of the norm problem for elementary operators acting on the hilbert schmidt class

semisimple orbits of lie algebras and card shuffling measures on coxeter groups abstract random walk on the chambers of hyperplane arrangements is used to define a family of card shuffling measures h w x for a finite coxeter group w and real x 0 by algebraic group theory there is a map from the semisimple orbits of the adjoint action of a finite group of lie type on its lie algebra to the conjugacy classes of the weyl group choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the weyl group for types a b and the identity conjugacy class of w for all types it is proved that for q very good this measure on conjugacy classes is equal to the measure arising from h w q the possibility of refining to a map to elements of the weyl group is discussed

cycle structure of riffle shuffles a class of models for rife shufes f shufes related to certain expansive mappings of the unit interval is studied the main result concerns the cycle structure of the resulting random permutations in sn when n is large it describes the asymptotic distribution of the number of cycles of a given length relating this distribution to dynamical properties of the associated mapping this result generalizes a recent result of diaconis mcgrath and pitman 1 introduction the cycle structure of a random permutation chosen from the uniform distribution on the permutation group sn is reasonably well understood when n 1 the joint distribution of the large cycles is governed by poisson dirichlet asymptotics see 7 and 9 and the number of short cycles of a given length j is approximately poisson recently diaconis mcgrath and pitman 3 obtained analogous results for random permutations from a class of nonuniform distributions the so called a shufes these are random permutations obtained by cutting a deck of n cards into a packets and then randomly rifing the packets together using a bijection discovered by gessel 4 they obtained an exact formula for the joint distribution of the numbers nj j d 1 2 n of cycles of length j and used this formula to describe the asymptotics of both the large cycles and the small cycles as n 1 the behavior of the large cycles is governed by the same poisson dirichlet asymptotics that apply for random permutations from the uniform distribution the asymptotic behavior of the short cycles however is markedly different than that of uniform random permutations for large n and xed j the distribution of nj is approximately the negative binomial

affine shuffles shuffles with cuts the whitehouse module and patience sorting abstract type a affine shuffles are compared with riffle shuffles followed by a cut although these probability measures on the symmetric group s n are different they both satisfy a convolution property strong evidence is given that when the underlying parameter q satisfies gcd n q 1 1 the induced measures on conjugacy classes of the symmetric group coincide this gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents using representation theoretic work on the whitehouse module a formula is obtained for the cycle structure of a riffle shuffle followed by a cut it is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness generating functions for the first pile size in patience sorting from decks with repeated values are derived this relates to random matrices

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

variance of operators and derivations abstract the variance of a bounded linear operator a on a hilbert space h at a unit vector is defined by d a a 2 a 2 we show that two operators a and b have the same variance at all vectors h if and only if there exist scalars c with 1 such that b a 1 or a is normal and b a 1 further if a is normal then the inequality d b d a holds for some constant and all unit vectors if and only if b f a for a lipschitz function f on the spectrum of a variants of these results for c algebras are also proved where vectors are replaced by pure states we also study the related inequalities b x x b a x x a supposed to hold for all x b h or for all x b h n and all n n we consider the connection between such inequalities and the range inclusion d b b h d a b h where d a and d b are the derivations on b h induced by a and b if a is subnormal we study these conditions in particular in the case when b is of the form b f a for a function f

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

a topology for operator modules overw algebras abstract given a von neumann algebraron a hilbert space h the so calledr topology is introduced into b h which is weaker than the norm and stronger than the ultrastrong operator topology a rightr submodulexof b h is closed in ther topology if and only if for eachb b h the right ideal consisting of alla rsuch thatba x is weak closed inr equivalently xis closed in ther topology if and only if for eachb b h and each orthogonal family of projectionseiinrwith the sum 1 the conditionbei xfor alliimplies thatb x

semisimple conjugacy classes and classes in the weyl group abstract we discuss a map from the semisimple conjugacy classes of a finite group gf of lie type to the f conjugacy classes of its weyl group we obtain two expressions for the number of semisimple classes mapped by into a given f conjugacy class of w the first involves distinguished coset representatives in the affine weyl group and the second is the number of elements in the coroot lattice satisfying certain conditions the brauer complex plays a key role in the proof the map has recently proved of interest in connection with probabilistic and combinatorial group theory

some remarks on kasparov theory on considere une autre preuve de l invariance d homotopie du bifoncteur ext on etudie l associativite du produit de kasparov a travers la motion de connextion

the haagerup norm on the tensor product of operator modules it is proved that the analogy of the haagerup norm on the tenser product of submodules of b h over a von neumann algebra t b h is injective if r s b h are von neumann algebras with s injective and t r s then the natural map froms ts s equipped with the haagerup norm to cb r s the space of all completely bounded maps from r to s is shown to be an isometry and from this we deduce the result of chattejee and smith that the natural map from the central haagerup tenser product r c r to cb r r is an isometry for each von neumann algebra r it is also shown that for an elementary operator on a prime c algebra with zero socle or on a continuous von neumann algebra the norm is equal to the completely bounded norm

the apostol fialkow formula for elementary operators on banach spaces abstract let a a 1 a n and b 1 b n be n tuples of bounded linear operators on a banach space e the corresponding elementary operator e a b is the map s n i 1 a i sb i on l e and e a b denotes the induced operator s n i 1 a i sb i on the calkin algebra c e l e k e here t t k e for t l e we establish that if e has a 1 unconditional basis then dist e a b w c e e a b dist e a b w l e for all elementary operators e a b on l e where w stands for the weakly compact operators there is equality throughout if e l p 1 p canad j math 38 1986 1485 1524 which they proved for e l 2 using the non commutative weyl von neumann theorem due to voiculescu by contrast our arguments are based on subsequence techniques from banach space theory as a byproduct we obtain a positive answer to the generalized fong sourour conjecture for a large class of banach spaces we also explicitly compute the norm of the generalized derivation s as sb on c l 2 this improves a result due to fong and show that the resulting formula fails to hold on c l p

fixed points of normal completely positive maps on b h abstract given a sequence of bounded operators a j on a hilbert space h with j 1 a j a j 1 j 1 a j a j we study the map defined on b h by x j 1 a j x a j and its restriction to the hilbert schmidt class c 2 h in the case when the sum j 1 a j a j is norm convergent we show in particular that the operator 1 is not invertible if and only if the c algebra a generated by a j j 1 has an amenable trace this is used to show that may have fixed points in b h which are not in the commutant a of a even in the case when the weak closure of a is injective however if a is abelian then all fixed points of are in a even if the operators a j are not positive

elementary operators on antiliminal c algebras we establish the equivalence of the following three properties of a c algebra a a every positive elementary operator on a is completely positive b the norm and the cb norm coincide for every elementary operator on a c a is an extension of an antiliminal c algebra by an abelian one

k theoretic amenability for sl2 qp and the action on the associated tree abstract it is proved that if a locally compact group g acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable then g is k amenable in particular the canonical map from the full c algebra onto the reduced c algebra of g induces isomorphisms in k theory the main corollary of our result is that sl 2 q p and some other groups over local fields are k amenable

the central haagerup tensor product and maps between von neumann algebras abstract we introduce the central haagerup tensor product r f h r for a von neumann algebra r and we show that the natural injection into tbe space cb r r of completely bounded maps on r is isometric this is used to study mappings between von neumann algebras and to give a new characterization of injectivity this tensor product is also studied for c algebras a and we show that the injection into cb a a need not always be isometric depending on the topology of the primitive ideal space of a

on the first integrals of n th order autonomous systems the first integrals of n th order autonomous systems are considered by getting integrating factors using n 1 n 1 first integrals of the systems a necessary condition on the existence of global first integrals of n th order autonomous systems is presented it is also proved that n 1 n 1 functionally independent first integrals of an n th order autonomous system can be obtained if the system possesses n 1 n 1 functionally independent integrating factors based on one parameter lie groups admitted by n th order autonomous systems several methods to obtain first integrals of the systems are presented simultaneously several related examples are given to illustrate the feasibility and the effectiveness of the proposed method

differential systems the mapping over period for which is represented by a product of three exponential matrixes for some periodic differential systems the sufficient conditions of a representation of the mapping over period as a product of a three exponential matrix of the special aspect are obtained obtained results are used for research of problems of the existence and stability of periodic solutions of nonlinear systems and the asymptotic orbital stability of cycles of autonomous differential systems

plethystic formulas for permutation enumeration abstract we prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions and the permutation statistics we consider include the descent number peak number left peak number and the number of up down runs we apply these results to cyclic permutations involutions and derangements and more generally to derive formulas for counting all permutations by the above statistics jointly with the number of fixed points and jointly with cycle type a number of known formulas are recovered as special cases of our results including formulas of desarmenien foata gessel reutenauer stembridge fulman petersen diaconis fulman holmes zhuang and athanasiadis

duality and normal parts of operator modules for an operator bimodule x over von neumann algebras a b h and b b k the space of all completely bounded a b bimodule maps from x into b k h is the bimodule dual of x basic duality theory is developed with a particular attention to the haagerup tensor product over von neumann algebras to x a normal operator bimodule xn is associated so that completely bounded a b bimodule maps from x into normal operator bimodules factorize uniquely through xn a construction of xn in terms of biduals of x a and b is presented various operator bimodule structures are considered on a banach bimodule admitting a normal such structure

a factorization problem for normal completely bounded mappings abstract given an operator space x and a von neumann algebra a we consider a contractive mapping q a eh x eh a ncb x a formally defined by q aj xjk bk xjk ajbk from the extended haagerup tensor product a eh x eh a into the space of w continuous completely bounded maps from x into a we characterize elements of the range space im q by a factorization property involving decomposable operators and investigate various properties of that space in the case when x b is the predual of a von neumann algebra b im q is included in the space dec b a of decomposable operators from b into a regarding q as having values in that space we show that q is a quotient map onto its range then we prove that dec b a is a normal dual operator a bimodule and that im q dec b a is a strong operator a bimodule

identifying assembly maps in k and l theory in this paper we prove the equivalence of various algebraically or geometrically defined assembly maps used in formulating the main conjectures in k and l theory and c theory

random walks and hyperplane arrangements let be the set of chambers of a real hyperplane arrangement we study a random walk on introduced by bidigare hanlon and rockmore this includes various shuffling schemes used in computer science biology and card games it also includes random walks on zonotopes and zonotopal tilings we find the stationary distributions of these markov chains give good bounds on the rate of convergence to stationarity and prove that the transition matrices are diagonalizable the results are extended to oriented matroids

on the first integral and equivalence of nonlinear differential equations in this paper we study the equivalence of several different types of differential systems and provide a new method to find out the first integral of such differential systems we draw some sufficient conditions for a general time varying nonlinear differential system to be equivalent to a given differential system we use the obtained results to discuss the qualitative behavior of the periodic solutions of such time varying differential systems and get some interesting conclusions