Showing papers on "Symmetric group published in 1988"

Hecke algebras of type An and subfactors.

Abstract: In his paper [J-l] V. Jones introduced an index, which 'measures' the size of a subfactor in a II1 factor. The main result of that paper is that the index of a subfactor has to be either greater or equal than 4 or it has to be equal to 4cosZ(x//) for some l~N, I>3 and that there exist subfactors for all these index values. Similarly as for subgroups, the index alone does not characterize the subfactor up to conjugacy by automorphisms. The fact that there are only countably many possible index values < 4 seems to be related to another invariant. Subfactors with index less than 4 always have trivial centralizers, (or relative commutants), i.e. the only elements of the factor which commute with every element of the subfactor are multiples of the identity. On the other hand, the examples given in I-J-l] for subfactors with index greater than 4 all have nontrivial centralizers. Furthermore, all known examples of subfactors with trivial relative commutants have as index an algebraic integer. At the current state of knowledge, it is still unknown whether there are only countably many values possible for the index of subfactors with trivial centralizers. Note however, that the set of all possible index values of a subfactor with trivial centralizer in an arbitrary II~ factor has to be a closed subset of R (see [HW]). Our original motivation for this paper was to study how subfactors of the hyperfinite II1 factor can be constructed via AF algebras. We provide a method of computing the index and we give an upper bound for the size of the centralizer of the constructed subfactor. Our general results will then be applied to the series of complex Hecke algebras H,(q), n~N of type A,_I. Their standard generators gx, g2, -.., gn1 satisfy the same relations as a set of simple reflections of the symmetric group S, except that the reflection property g~ = 1 is replaced by g ~ = ( q 1 ) g i + q . It is well-known that H,(q) is isomorphic to C S , if q is not a root of unity. If the parameter is a root of unity, Hn(q) may no longer bc sernisimple and its structure is not known in general. This is, however, the most interesting case as far as subfactors are concerned. We define representations p of Ha(q) such that p(H,(q)) is semisimple for all n~N. Together with

On the O'Nan-Scott theorem for finite primitive permutation groups

Abstract: We give a self-contained proof of the O'Nan-Scott Theorem for finite primitive permutation groups.

The automorphism group of a shift of finite type

Abstract: Let (XT,AT) be a shift of finite type, and G = aut(vT) denote the group of homeomorphisms of XT commuting with ¢T. We investigate the algebraic properties of the countable group G and the dynamics of its action on XT and associated spaces. Using "marker" constructions, we show G contains many groups, such as the free group on two generators. However, G is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of CRT whose exact value is not known. We prove that, modulo a few points of low period, G acts transitively on the set of points with least aT-period n. Using padic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of G on the dimension group of aT iS investigated. We show there are no proper infinite compact G-invariant sets. We give a complete characterization of the G-orbit closure of a continuous probability measure, and deduce that the only continuous G-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.

Combinatorics, Bethe Ansatz, and representations of the symmetric group

Abstract: Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations)

Generalized quotients in Coxeter groups

Abstract: For (W, S) a Coxeter group, we study sets of the form W/V = {w E W I l(wv) = 1(w) + I(v) for all v E V}, where V C W. Such sets W/V, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when V C S (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in W/V, for general V C W, are lexicographically shellable. The Mobius function on W/V under Bruhat order takes values in {-1, 0, +1}. For finite groups W, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that W/V is always a complete meet-semilattice and a convex order ideal as a subset of W under weak order. Descent classes DI = {w E W Il1(ws) < 1(w) Xt s e I, for all s E S}, I C S, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.

Covering Problems for Markov Chains

Abstract: Upper and lower bounds are given on the moment generating function of the time taken by a Markov chain to visit at least $n$ of $N$ selected subsets of its state space. An example considered is the class of random walks on the symmetric group that are constant on conjugacy classes. Application of the bounds yields, for example, the asymptotic distribution of the time taken to see all $N!$ arrangements of $N$ cards as $N\rightarrow\infty$ for certain shuffling schemes.

The Ubiquitous Young Tableau

Abstract: Young tableaux have found extensive application in combinatorics [Vie 84], group representations [Jam 78], invariant theory [DRS 74, DKR 78], symmetric funcions [Mad 79], and the theory of algorithms [Knu 73, pages 48-72]. This paper is an expository treatment of some of the highlights of tableaux theory. These include the hook and determinantal formulae for enumeration of both standard and generalized tableaux, their connection with irreducible representations of matrix groups, and the Robinson-Schensted-Knuth algorithm. 1 Three families of tableaux Young tableaux were first introduced in 1901 by the Reverend Alfred Young [You 01, page 133] as a tool for invariant theory. Subsequently, he showed that they can give information about representations of symmetric groups. Since then, tableaux have played an important role in many areas of mathematics from enumerative combinatorics to algebraic geometry. This paper is a survey of some of these applications. In recent years the number of tableaux of various types has been increasing at an impressive rate. To limit this paper to a reasonable length, our discussion will be restricted to three fundamental families of tableaux: ordinary, shifted and oscillating. The rest of this section will be devoted to the definitions and notation need to describe these arrays. In Section 2 we present the hook and determinantal formulae for enumeration of standard tableaux. The third section examines the connection with representations of the symmetric group. The Robinson-Schensted algorithm appears in Section 4 as a combinatorial way of explaining the decomposition of the regular representation. The next four sections rework the material from the first four using generalized tableaux (those with repeated entries), representations of general linear and symplectic groups, and the theory of symmetric functions. Section 9 is a brief exposition of some open problems. 1.1 Ordinary tableaux In what follows, N and P stand for the non-negative and positive integers respectively. A partition λ of n ∈ N, written λ ` n, is a sequence of positive integers λ = (λ1, λ2, · · · , λl) in weakly decreasing order such that ∑l i=1 λi = n. The λi are called the parts of λ. The unique partition of 0 is λ = φ. The shape of λ is an array of boxes (or dots or cells) with l left-justified rows and λi boxes in row i. We will use λ to represent both the partition and its shape, while (i, j) will denote the cell in row i and column j. By way of illustration, the following figure shows the shape of the partition λ = (2, 2, 1) ` 5 with cell (3,1) displayed as a diamond. In deference to Alfred Young’s nationality, we have chosen to draw partition shapes in the English style, i.e., as if they were part of a matrix. The reader should be aware that some mathematicians (notably the French) prefer to use the conventions of coordinate geometry where λ1 cells are placed along the x-axis, λ2 cells are placed along the line y = 1, etc. To them and to Rene Descartes, we abjectly apologize.

Decomposition matrices for spin characters of symmetric groups

Abstract: Methods are developed for determining the decomposition matrices for the spin characters of the symmetric groups Sn for an odd prime p. Some general results are obtained which are non-trivial modifications of the corresponding results for ordinary characters. The methods are used to determine the decomposition matrices for 3 ≦ n ≦ ll, and p = 3 but with an interesting ambiguity in the case n = 9. The second author will deal separately with the cases p = 5, 7, 11.

Matrix elements of the third-order spin-adapted reduced Hamiltonian.

Abstract: A formalism recently developed to obtain a matrix representation of the second-order spin-adapted reduced Hamiltonian (J. Karwowski, W. Duch, and C. Valdemoro, Phys. Rev. A 33, 2254 (1986)) and derived from the symmetric-group approach to configuration-interaction methods has been generalized to the case of the third-order spin-adapted reduced Hamiltonian. As in the second-order case, the matrix elements are linear combinations of generalized two-electron integrals with coefficients explicitly related to traces of products of the orbital occupation-number operators. The third-order spin-adapted reduced Hamiltonian matrix has been block diagonalized according to the irreducible representations of the symmetric group S/sub 3/.

Graphs whose full automorphism group is a symmetric group

Abstract: We address the problem of describing all graphs Γ such that Aut Γ is a symmetric group, subject to certain restrictions on the sizes of the orbits of Aut Γ on vertices. As a corollary of our general results, we obtain a classification of all graphs Γ on v vertices with Aut Γ ≅ Sn, where ν < min{5n, ½n(n – 1)}.

The structure of the centralizer of a permutation

Abstract: Relative to the symmetric groups Sn the structure of centralizers of permutations are known as direct products of certain general wreath products. A recent generalization of the cycle notation for partial one-one transformations (charts) is applied to show that relative to the symmetric inverse semigroups Cn the structure of centralizers of permutations are also direct products of certain subsemigroups of a wreath product, and this latter wreath product includes the former as a subgroup. A necessary and sufficient condition is given for two charts to commute and the approach for the Cn-case parallels and generalizes the one for the Sn-case. As a result, the Cn-case yields the standard known characterizations of commuting permutations, as well as formulas for the orders of centralizers as corollaries. It is an open problem to extend these results to the centralizers of arbitrary charts in Cn.

Electronic properties of disordered materials: a symmetric group approach

Abstract: The presence of disorder in a solid invalidates most of the methods developed for computing electronic properties. There is a need for a theory which does not rely on an ansatz concerning the wavefunctions and which does not present its solution in terms of impossible to perform summations of infinite and sometimes unidentified perturbation series. In an earlier set of papers, the authors have shown how the transfer matrix method can be generalised in the one-dimensional (1D) case to treat the case of a disordered solid, enabling densities of states and mean inverse localisation lengths to be calculated. This was done by application of the symmetric group to direct products of transfer matrices. Here the authors consider the case of arbitrary dimensions and show that, whereas the completely symmetric representation sufficed to solve the problem in 1D, in higher dimensions a more extended application of group theory is required. They derive an expression for the density of states in a secular-equation-like form involving diagonalising an infinite matrix which can in practice be truncated to finite dimensions. In the limit of an infinite matrix the theory is exact, and comparisons of calculations made using their theory with simulations for finite systems show that the authors can obtain highly accurate results with a minimum of computational effort.

The Demazure–Tits subgroup of a simple Lie group

Abstract: The Demazure–Tits subgroup of a simple Lie group G is the group of invariance of Clebsch–Gordan coefficients tables (assuming an appropriate choice of basis). The structure of the Demazure–Tits subgroups of An, Bn, Cn, Dn, and G2 is described. Orbits of the permutation action of the DT group in any irreducible finite‐dimensional representation space of A2, C2, and G2 are decomposed into the sum of irreducible representations of the DT group.