Showing papers on "Symmetric group published in 1994"

Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces

Abstract: In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.

Permutation Statistics of Indexed Permutations

Abstract: The definitions of descent, excedance, major index, inversion index and Denert's statistics for the elements of the symmetric group Ld are generalized to indexed permutation, i.e. the elements of the group Snd: = Zn ? Ld, where ? is the wreath product with respect to the usual action of Ld by permutation of {1, 2,?, d}.It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomial, analogous to the Eulerian polynomial, is computed as the f-eulerian polynomial of a simple polynomial. The descent polynomial is shown to equal the h-polynomial (essentially the h -vector) of a certain triangulation of the unit d-cube. This is proved by a bijection which exploits the fact that the h-vector of the simplicial complex arising from the triangulation can be computed via a shelling of the complex. The famous formula ?d ? 0Edxd /d! = sec x + tan x, where Ed is the number of alternating permutations in Ld, is generalized in two different ways, one relating to recent work of V. I. Arnold on Morse theory. The major index and inversion index are shown to be equidistributed over Snd . Likewise, the pair of statistics (d, maj) is shown to be equidistributed with the pair (?, den), where den is Denert's statistic and ? is an alternative definition of excedance. A result relating the number of permutations with k descents to the volume of a certain 'slice' of the unit d-cube is also generalized.

Subgroup Lattices and Symmetric Functions

Abstract: Introduction Subgroups of finite Abelian groups Hall-Littlewood symmetric functions Some enumerative combinatorics Some algebraic combinatorics.

Investigations in Algebraic Theory of Combinatorial Objects

Abstract: Series Editor's Preface. Preface to the English Edition. Preface to the Russian Edition. Part 1: Cellular Rings. 1.1. Cellular Rings and Groups of Automorphisms of Graphs I.A. Faradzev, M.H. Klin, M.H. Muzichuk. 1.2 On p-Local Analysis of Permutation Groups V.A. Ustimenko. 1.3. Amorphic Cellular Rings Ja. Ju. Gol'fand, A.V. Ivanov, M.H. Klin. 1.4 The Subschemes of the Hamming Scheme M.E. Muzichuk. 1.5. A Description of Subrings in V(Sp1 x Sp2 x ... x Spm) Ja. Ju. Gol'fand. 1.6. Cellular Subrings of the Symmetric Square of a Cellular Ring of Rank 3 I.A. Faradzev. 1.7. The Intersection Numbers of the Hecke Algebras H(PGLn(q),BWjB) V.A. Ustimenko. 1.8. Ranks and Subdegrees of the Symmetric Groups Acting on Partitions I.A. Faradzev, A.V. Ivanov. 1.9. Computation of Lengths of Orbits of a Subgroup in a Transitive Permutation Group A.A. Ivanov. Part 2: Distance-Transitive Graphs. 2.1. Distance-Transitive Graphs and Their Classification A.A. Ivanov. 2.2. On Some Local Characteristics of Distance-Transitive Graphs A.V. Ivanov. 2.3. Action of the Group M12 on Hadamard Matrices I.V. Chuvaeva, A.A. Ivanov. 2.4. Construction of an Automorphic Graph on 280 Vertices Using Finite Geometries F.L. Tchuda. Part 3: Amalgams and Diagram Geometries. 3.1. Applications of Group Amalgams to Algebraic Graph Theory A.A. Ivanov, S.V. Shpectorov. 3.2. A Geometric Characterization of the Group M22 S.V. Shpectorov. 3.3. Bi-Primitive Cubic Graphs M.E. Lofinova, A.A. Ivanov. 3.4. On Some Properties of Geometries of Chevalley Groups and Their Generalizations V.A. Ustimenko. Subject Index.

On the Kronecker product of Schur functions of two row shapes

Abstract: The Kronecker product of two homogeneous symmetric polynomials P1 and P2 is dened by means of the Frobenius map by the formula P1P2 = F (F 1 P1)(F 1 P2) When P1 and P2 are Schur functions s and s respectively, then the resulting product s s is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams and Taking the scalar product of ss with a third Schur function s gives the so-called Kronecker coecient g =hss;si which gives the multiplicity of the representation corresponding to in the tensor product In this paper, we prove a number of results about the coecients g when both and are partitions with only two parts, or partitions whose largest part is of size two We derive an explicit formula for g and give its maximum value

Galois rigidity of pure sphere braid groups and profinite calculus

Abstract: Let C be a class of finite groups closed under the for- mation of subgroups, quotients, and group extensions. For an algebraic variety X over a number field k, let π C 1 (X) denote the (C-modified) profi- nite fundamental group of X having the absolute Galois group Gal( ¯ k/k) as a quotient with kernel π C (X¯ ) the maximal pro-C quotient of the geo- metric fundamental group of X. The purpose of this paper is to show certain rigidity properties of π C (X) for X of hyperbolic type through the study of outer automorphism group Outπ C (X )o fπ C 1 (X). In particular,

Symmetric Groups and Quasi-Hereditary Algebras

Abstract: Quasi-hereditary algebras were introduced by L. Scott [S] in order to study highest weight categories arising in the representation theory of semisimple complex Lie algebras and algebraic groups, and important results were proved by Cline, Parshall and Scott (see [CPS1,2]). These algebras can be defined entirely in ring-theoretic terms; and they were studied from this point of view by Dlab and Ringel (see [DR1,2], [R1,2]) and by others. In particular it turns out that quasi-hereditary algebras are quite common.

On the conjugacy of element-conjugate homomorphisms

Abstract: A groupG isacceptable if a homomorphism ϕ from a finite group Γ toG is determined up to conjugation by the conjugacy classes of the elements ϕ(γ). Some progress is made toward classifying acceptable Lie groups.

On restrictions of Irreducible Modular Representations of Semisimple Algebraic Groups and Symmetric Groups to Some Natural Subgroups, I

Abstract: (1994). On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II. Communications in Algebra: Vol. 22, No. 15, pp. 6175-6208.

Near embeddings of hypercubes into Cayley graphs on the symmetric group

Abstract: Simulations of hypercube networks by certain Cayley graphs on the symmetric group are investigated. Let Q(k) be the familiar k-dimensional hypercube, and let S(n) be the star network of dimension n defined as follows. The vertices of S(n) are the elements of the symmetric group of degree n, two vertices x and y being adjacent if xo(1,i)=y for some i. That is, xy is an edge if x and y are related by a transposition involving some fixed symbol (which we take to be /spl I.bold/1). This network has nice symmetry properties, and its degree and diameter are sublogarithmic as functions of the number of vertices, making it compare favorably with the hypercube network. These advantages of S(n) motivate the study of how well it can simulate other parallel computation networks, in particular, the hypercube. The first step in such a simulation is the construction of a one-to-one map f:Q(k)/spl rarr/S(n) of dilation d, for d small. That is, one wants a map f such that images of adjacent points are at most distance d apart in S(n). An alternative approach, best applicable when one-to-one maps are difficult or impossible to find, is the construction of a one-to-many map g of dilation d, defined as follows. For each point x/spl isin/Q(k), there is an associated subset g(x)/spl sube/V(S(n)) such that for each edge xy in Q(k), every x'/spl isin/g(x) is at most distance d in S(n) from some y'/spl isin/g(y). Such one-to-many maps allow one to achieve the low interprocessor communication time desired in the usual one-to-one embedding underlying a simulation. This is done by capturing the local structure of Q(k) inside of S(n) (via the one-to-many embedding) when the global structure cannot be so captured. Our results are the following. 1) There exist the following one-to-many embeddings: a) f:Q(k)/spl rarr/S(3k+1) with dilation (f)=1; b) f:Q(11k+2)/spl rarr/S(13k+2) with dilation (f)=2. 2) There exists a one-to-one embedding f:Q(n2/sup n/spl minus/1/)/spl rarr/S(2/sup n/) with dilation (f)=3. >

Finite distance-transitive generalized polygons

Abstract: Using the classification of the finite simple groups, we classify all finite generalized polygons having an automorphism group acting distance-transitively on the set of points. This proves an old conjecture of J. Tits saying that every group with an irreducible rank 2 BN-pair arises from a group of Lie type.

Sn-normal semigroups

Abstract: Certain subsemigroups of the full transformation semigroup Tn on a finite set of cardinality n are investigated, namely those subsemigroups S of Tn, which are normalised by the symmetric group on n elements, the group of units of Tn. The Sn-normal closure of an element of Tn is determined, and the structure of the Sn-normal ideals consisting of the members of Tn whose image contains at most r elements is studied.

The maximum order of finite groups of outer automorphisms of free groups

Abstract: Let Fr be free group of rank r > l and O u t F r = A u t Fr/InnFr, the outer au tomorph ism group (automorphisms modulo inner automorphisms). We shall prove the following. Theorem. The maximum order of a f ini te subgroup G of Out F, is 12, for r= 2, and 2 r r !, for r > 2. Furthermore the f inite subgroup of Out F, realizing the maximum order is unique up to conjugacy, for r > 3. The proof of the above theorem is based on the following geometric realization result which was observed first in [Z, p. 478], see also [Cu]. Proposition 0 Let G be a .finite subgroup of Out F,. Then there exist a f inite connected graph F with ~ 1 F = F , and an action of G on F realizing the given action on F,. Remarks and definitions. Let F' be a subgraph obtained by deleting all free edges of F, i.e., edges with vertex of valence 1. Note that the G action restricts on the new graph F ' and the fundamental g roup and the induced action on it do not change. So we may assume in Proposi t ion 0 that F contains no vertex of valence 1. We use Sym F to denote the symmetry group of F (in a combina tor ial sense, i.e., mapping vertices to vertices and oriented edges to oriented edges, or equivalently, homeomorph i sms acting linearly on edges). In general we allow inversions of edges, i.e., reflections in the midpoints of edges (which is not considered as the identity). Then, in Proposi t ion 0, we may also assume that F has no vertices of valence 2 by amalgamat ing the two adjacent edges containing the same vertex of valence 2 into one edge, again G acts on the new graph with the same induced action on the fundamenta l group. We say that a g roup G acts effectively on F if it can be embedded into Sym F, or equivalently, no non-trivial element of G is the identity on F. For a connected graph F, its rank, denoted by rank F, is the rank of its fundamental group. Lemma 1 Let F be a connected graph of rank r > 1 without vertices of valence 1 and G be a f ini te group acting on F. Then G acts effectively on F if and only if the induced action on ~ F injects into Out ~ F.

Eulerian Calculus, I

Abstract: Further transformations are derived on the symmetric group and related structures that extend previous works on the equidistributions of several classical univariable statistics.

On an integral representation for the genus series for 2-cell embeddings

Abstract: An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, CO,, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals. 1. A POWER SERIES REPRESENTATION FOR THE GENUS SERIES A map is a 2-cell embedding of a connected unlabelled graph ', with loops and multiple edges allowed, in a closed surface X, without boundary, which is assumed throughout to be oriented. The deletion of ' separates X into regions homeomorphic to open discs, called the faces of the map, and the number of edges bordering a face is called its degree. A map is rooted by distinguishing a mutually incident vertex, edge and face. The genus series for a class of maps is the formal generating series for the number of inequivalent maps with respect to genus, and the numbers of vertices, edges and faces. It is assumed hereinafter that maps are rooted. The general approach adopted here combines ideas of Jackson and Visentin [11] with those of 't Hooft [8] and Bessis, Itzykson and Zuber [3] who, in the above terminology, derived the genus series for diagrams akin to a class of maps by techniques from conformal field theory. Although [81 and [3] are important papers, they have remained largely inaccessible to combinatorialists because of their uncertainty about the automorphisms of these diagrams as combinatorial structures. In this paper, an explicit construction is given for an integral representation for the genus series for general maps directly from the combinatorial axiomatisation for embeddings on oriented surfaces. Moreover, we also develop methods which are extensible to vertex-regular maps (vertices have the same degree) and thence, by restriction, to quadrangulations (maps whose faces are bounded by four edges). This is done by examining dipoles (maps with two vertices) in detail. Although the argument is an algebraic one, based on the ring of formal power series, to assert that particular series belong to the ring, it is necessary to Received by the editors July 1, 1992 and, in revised form, August 15, 1993. 1991 Mathematics Subject Classification. Primary 05A1 5, 20C1 5; Secondary 57N37. (?) 1994 American Mathematical Society 0002-9947/94 $1.00 + S.25 per page

Automorphisms and Isomorphisms of Symmetric and Affine Designs

Abstract: Given a finite group G, for all sufficiently large d and for each q > 3 there are symmetric designs and affine designs having the same parameters as PG(d, q) and AG(d, q), respectively, and having full automorphism group isomorphic to G.

Strings and Two-dimensional QCD for Finite N

Abstract: The string theory description of SU ( N ) Yang-Mills on an arbitrary two-dimensional manifold, previously developed for the large N asymptotic expansion, is extended to include finite values of N . The theory is considered from two points of view, first using a canonical Hamiltonian formulation, second using a global description of the partition function. In both formalisms, the effect on the string theory of taking a finite value of N is described by a local projection operator which has a simple description in terms of the symmetric group S n .

Partition identities and labels for some modular characters

Abstract: In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups Sn, of the finite symmetric groups S,n in characteristic 5 . One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. 0. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of Sn, in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and B * below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture B *, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels. In ? 1 we give a brief description of the groups Sn and their representations, leading up to Conjectures A and B * as they were formulated in [BMO]. That section also presents the background for Conjecture B as stated in [A3] and the equivalence of Conjectures B and B * is explained. Sections 2 and 3 are devoted to the proof of Conjecture B, and ?4 to the proof of Conjecture A. 1. THE CONJECTURES AND THEIR BACKGROUND For facts concerning the general representation theory of finite groups needed in the following, the reader is referred to [F, NT]. In 191 1 Schur [S1] proved that the finite symmetric groups S, have covering groups S, of order 2IS, I = 2 * n ! This means that there is an exact sequence