Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

Brauer Diagrams, Updown Tableaux and Nilpotent Matrices

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg's result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all where is a complete flag in is a nondegenerate alternating bilinear form on and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both and instead of Steinberg's variety of where are two complete flags in and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both .     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

A Geometric Characterization of Fischer's Baby Monster

The sporadic simple group F ₂ known as Fischer's Baby Monster acts flag-transitively on a rank 5 P-geometry . P-geometries are geometries with string diagrams, all of whose nonempty edges except one are projective planes of order 2 and one terminal edge is the geometry of the Petersen graph. Let be a flag-transitive P-geometry of rank 5. Suppose that each proper residue of is isomorphic to the corresponding residue in . We show that in this case is isomorphic to . This result realizes a step in classification of the flag-transitive P-geometries and also plays an important role in the characterization of the Fischer–Griess Monster in terms of its 2-local parabolic geometry.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

On Distance-Regular Graphs with Height Two, II

Let be a distance-regular graph with diameter and height , where . Suppose that for every in and every in , the induced subgraph on is isomorphic to a complete multipartite graph with . Then and is isomorphic to the Johnson graph .     

Profinite and Continuous Higher K-Theory of Exact Categories, Orders, and Group Rings

Let be a rational prime, an exact category. In this article, we define and study for all , the profinite higher K-theory of , that is as well as , where is the -dimensional mod- Moore space. We study connections between and prove several -completeness results involving these and associated groups including the cases where is the category of finitely generated (resp. finitely generated projective) modules over orders in semi-simple algebras over number fields and p-adic fields. We also define and study continuous K-theory of orders in p-adic semi-simple algebras and show some connection between the profinite and continuous K-theory of .     

Dynamic ℤ d -Random Walks in a Random Scenery: A Strong Law of Large Numbers

In this paper, we study a _d -random walk on nearest neighbours with transition probabilities generated by a dynamical system . We prove, at first, that under some hypotheses, verifies a local limit theorem. Then, we study these walks in a random scenery , a sequence of independent, identically distributed and centred random variables and show that for certain dynamic random walks, satisfies a strong law of large numbers.     

A Constructive Proof of the Generalized Gelfand Isomorphism

Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius -homomorphism. For , this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let X be a compact Hausdorff space, the th symmetric power of X, and the algebra of continuous complex-valued functions on X with the sup-norm; then the evaluation map defined by the formula identifies the space with the space of all Frobenius -homomorphisms of the algebra into with the weak topology.     

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

In this paper, we investigate self-dual codes over finite rings, specifically the ring of integers modulo 2m. Type II codes over are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over are also studied.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes with are either bipartite or antipodal, i.e., intersection numbers satisfy either for all for all . In this paper, we show that imprimitive -polynomial association schemes with are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either .     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

A Completion of \mathbb{Z} is a Field

We define various ring sequential convergences on and . We describe their properties and properties of their convergence completions. In particular, we define a convergence on by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields Further, we show that is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.     

Book Notices

Given the minimization problem of a real-valued function let A be any algorithm of type with that converges to a local minimum . In this note, new assumptions on f(x) under which A converges linearly to x* are established. These include the ones introduced in the literature which involve the uniform convexity of f(x).     

Homogeneity of a Distance-Regular Graph Which Supports a Spin Model

A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra (W). In this paper we study the distance-regular graphs whose Bose-Mesner algebra satisfies W (W). Suppose W has at least three distinct entries. We show that is 1-homogeneous and that the first and the last subconstituents of are strongly regular and distance-regular, respectively.     

Annihilators in Normal Autometrized Algebras

The concepts of an annihilator and a relative annihilator in an autometrized l-algebra are introduced. It is shown that every relative annihilator in a normal autometrized l-algebra is an ideal of and every principal ideal of is an annihilator of . The set of all annihilators of forms a complete lattice. The concept of an I-polar is introduced for every ideal I of . The set of all I-polars is a complete lattice which becomes a two-element chain provided I is prime. The I-polars are characterized as pseudocomplements in the lattice of all ideals of containing I.     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, , is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of on genus zero permutations and Baxter permutations. Our q-enumerative results relate the statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.