Showing papers on "Symmetric group published in 1995"

Homological stability for automorphism groups of free groups

Abstract: Let F, be a free group on n generators, Aut(F,) its group of automorphisms, and Out(F,) its outer automorphism group, the quotient of Aut(F,) by inner automorphisms. There has been much progress of late in the study of these groups via the one-dimensional model which arises from regarding F, as the fundamental group of a graph; see e.g., [BH] and [CV]. In this paper we return to a three-dimensional model first used by J. H. C. Whitehead in the 1930's, which involves looking at embedded 2-spheres in a connected sum of S 1 x S2's. Refining Whitehead's techniques and applying subsequent results of Laudenbach, we use this three-dimensional model to prove:

On the distribution of the number of cycles of elements in symmetric groups

Abstract: Abstract. We give a formula for the number of elements in a fixed conjugacy class of a symmetric group whose product with a cyclic permutation has a given number of cycles. A consequence is a very short proof of the formula for the number εg(n) of ways of obtaining a Riemann surface of given genus g by identifying in pairs the sides of a 2n-gon. This formula, originally proved by a considerably more difficult method in [1], was the key combinatorial fact needed there for the calculation of the Euler characteristic of the moduli space of curves of genus g. As a second application, we show that the number of ways of writing an even permutation π ∈ SN as a product of two N -cycles always lies between 2(N − 1)!/(N − r + 2) and 2(N − 1)!/(N − r + 19/29), where r is the number of fixed points of π, and that both constants “2” and “19/29” are best possible.

Unitary group approach to spin‐adapted open‐shell coupled cluster theory

Abstract: We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with nc doubly occupied and nv unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(nc) ⊗ U(nV) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(nc + nv). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group SM. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(nc) ⊗ U(ns) tensors, can be shown to be connected, the irreducible tensor operators of U(nc) ⊗ U(nv) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=nc + nv + ns is not simply reducible under the group U(nc) ⊗ U(ns) ⊗ U(nv). The multiplicity problem is resolved using the group chain U(n) ⊃ U(nc + nv) ⊗ U(ns) ⊃ U(nc) ⊗U(ns)⊗ U(nv) ⊗ U(nv). The labeling of the resulting configuration-state functions (which, in general, are not GT states when nc > 1) by the irreducible representations of the intermediate group U(nc + nv) ⊗U(ns) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.

Intertwining operators associated to the group

Abstract: For any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in G. There exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a singular set of parameter values (containing only certain negative rational numbers). This paper constructs an integral transform implementing the intertwining operator for the group S3, the symmetric group on three objects, for parameter value > ! . The transform is realized as an absolutely continuous measure on a compact subset of M2(R), which contains the group as a subset of its boundary. The construction of the integral formula involves integration over the unitary group U(3) . Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in G. It has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a "singular set" of parameter values. This singular set contains only negative values and is closely linked to the zero-set of the Poincare series of G. This paper constructs an integral transform implementing the intertwining map for the group S3, the symmetric group on three objects, for positive parameter values. Previously this had been done only for the group Z2 (acting by sign-change on R) where the transform is a classical fractional integral. The transform in this paper has its origin in the adjoint action of the unitary group U(3) on the linear space of real diagonal 3 x 3 matrices (the complexification of the maximal torus). This will lead to a transform realized as an absolutely continuous measure on a certain compact subset of M2(R). Here is a concise statement of the main result (rephrased from formulas (5.1), (5.6)): the operator V intertwines the differential-difference operators Received by the editors August 22, 1994. 1991 Mathematics Subject Classification. Primary 22E30, 33C80; Secondary, 33C50, 20B30.

Reduced decompositions in Weyl groups

Abstract: Let R be a root system with fixed basis ϵ and let W be its Weyl group. For every element w ϵ W , there exists a natural correspondence between reduced decompositions of w and some linear orders of the set of inversions of w . We study combinatorial properties of these orders. Furthermore, in the case of a symmetric group, for any permutation w we construct a linear action of the symmetric group ϵ l ( w ) on the space spanned by reduced decompositions of w .

The Homology of Partitions with an Even Number of Blocks

Abstract: Let \Pi_{2n}^e denote the subposet obtained by selecting even ranks in the partition lattice \Pi_{2n}. We show that the homology of \Pi_{2n}^e has dimension {{(2n)!}\over {2^{2n-1}}} E_{2n-1}, where E_{2n-1} is the tangent number. It is thus an integral multiple of both the Genocchi number and an Andre or simsun number. Using the general theory of rank-selected homology representations developed in l22r, we show that, for the special case of \Pi_{2n}^e, the character of the symmetric group S2n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers bi(n), 2 ≤ i ≤ n, defined recursively. We conjecture that, for the full automorphism group S2n, the homology is a sum of permutation modules induced from Young subgroups of the form S^i_2 \times S^{2n-2i}_1, with nonnegative integer multiplicity bi(n). The nonnegativity of the integers bi(n) would imply the existence of new refinements, into sums of powers of 2, of the tangent number and the Andre or simsun number an(2n). Similarly, the restriction of this homology module to S2n−1 yields a family of integers di(n), 1 ≤ i ≤ n − 1, such that the numbers 2−idi(n) refine the Genocchi number G2n. We conjecture that 2−idi(n) is a positive integer for all i. Finally, we present a recursive algorithm to generate a family of polynomials which encode the homology representations of the subposets obtained by selecting the top k ranks of \Pi_{2n}^e, 1 ≤ k ≤ n − 1. We conjecture that these are all permutation modules for S2n.

Finite quotients of the automorphism group of a free group

Abstract: The automorphism group AutFn of a free group Fn of rank n acts on the product of n copies of a group G by substituting n elements of G into the words defining an automorphism of the free group. This gives rise to an antihomomorphism from AutFnto a permutation group. We determine this antihomomorphic image of AutFn when G is the semidirect product Zp x Zq

Tensor products and dimensions of simple modules for symmetric groups

Abstract: LetK be an algebraically closed field of characteristic,p>0 and letDλ be the simple modules of the symmetric groupSr overK where λ is a p-regular partition ofr. The dimensions ofDλ for λ with at mostn parts are the same as the multiplicities of direct summands ofD⊗r whereE is the natural module for the groupGLn(K). Whenn=2 we determine generating functions for these multiplicities and hence for the dimensions ofDλ for all partitions λ with two parts. These can be expressed as rational functions of Chebyshev polynomials; and we obtain explicit formulae for the coefficients.

Finite Geometry, Dirac Groups and the Table of Real Clifford Algebras

Abstract: Associated with the real Clifford algebra Cl(p, q), p + q = n, is the finite Dirac group G(p, q) of order 2n+1. The quotient group V n = G(p, q)/ *#x007B;± 1*#x007D;, viewed additively, is an ndimensional vector space over GF(2) = *#x007B;0, 1*#x007D; which comes equipped with a quadratic form Q and associated alternating bilinear form B. Properties of the finite geometry over GF(2) of V n B, Q — in part familiar, in part less so — are given a rather full description, and a dictionary of translation into their Dirac group counterparts is provided. The knowledge gained is used, in conjunction with facts concerning representations of G(p, q), to give a pleasantly clean derivation of the well-known table of Porteous (1969/1981) of the algebras Cl(p, q). In particular the finite geometry highlights the “antisymmetry” of the table about the column p - q = -1. Several low-dimensional illustrations are given of the application of finite geometry results to Dirac groups. Particular emphasis is laid on certain interesting finite geometry symmetry methods, the latter being given a rather full treatment in the appendices. Finite geometry is also used to study the automorphisms of the Dirac groups, and the splitting of certain exact sequences.

Canonical bases for the Brauer centralizer algebra

Abstract: In this paper we construct canonical bases for the Birman-Wenzl algebra BWn, the q-analogue of the Brauer centralizer algebra, and so define left, right and two-sided cells. We describe these objects combinatorially (generalizing the Robinson-Schensted algorithm for the symmetric group) and show that each left cell carries an irreducible representation of BWn. In particular, we obtain canonical bases for each representation, defined over Z. The same technique generalizes to an arbitrary tangle algebra and Rmatrix [R]; in particular to centralizers of the quantum group action on V ⊗r, for V a finite dimensional representation of a quantum group. BWn occurs for particular values of the parameters (q, r, x) as the centralizers of the action of Uqsp2k or Uqok on the n-th tensor power of its standard representation V . One may presumably transfer the bases of the BWn modules to give a basis of representations occurring in V ⊗n (as in [GL]), and it is natural to conjecture that the basis so obtained coincides with that of [L,§27]. Of the Weyl groups, only in the symmetric group are the cell representations irreducible. In this respect BWn is similar to Sn. One would expect this because of the relation with quantum groups, which also behave like Hecke algebras of type A [L]. Moreover, our main new insight into the structure of BWn is precisely of this form—we show that every representation is induced from a representation of a symmetric group in a precise way (see §6.5). This paper is essentially self-contained, except for an appeal to the solution of the corresponding problem for Sn in [KL,1.4]. In particular, we make no further mention of quantum groups and use no previous work on the structure of BWn (e.g. [BW,HR,W]) except for its description as a

Distance-transitive representations of the sporadic groups

Abstract: A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free. In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.

A toral configuration space and regular semisimple conjugacy classes

Abstract: For any topological space X and integer n ≥ 1, denote by C n ( X ) the configuration space The symmetric group S n acts by permuting coordinates on C n ( X ) and we are concerned in this note with the induced graded representation of S n on the cohomology space H *( C n ( X )) = ⊕ i H i ( C n ( X ), ℂ), where H i denotes (singular or de Rham) cohomology. When X = ℂ, C n ( X ) is a K (π, 1) space, where π is the n -string pure braid group (cf. [3]). The corresponding representation of S n in this case was determined in [5], using the fact that C n ( C ) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

Orthogonal bases of symmetrized tensor spaces

Abstract: It is shown that a symmetrized tensor space does not have an orthogonal basis consisting of standard symmetrized tensors if the associated permutation group is 2-transitive. In particular, no such basis exists if the group is the symmetric group or the algernating group as conjectured by T.-Y. Tam and the author.

Group Theory: An Intuitive Approach

Abstract: The physical principles of group theory examples of groups groups as mathematical objects groups, combinations, subsets the group as a representation of itself properties of representations the symmetric group and its representations applications of symmetric group representations the rotation groups and their relatives representations of groups SO(3) and SU(2) applications of representations of SO(3) and O(3) Lie algebras representations of Lie algebras groups of crystals and molecules.

Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups G(r,p,n)

Abstract: Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike, Broue and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene, who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,p,n)$ given by Ariki and Koike.

A Note on the Number of $t$-Core Partitions

Abstract: A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a partition in the natural way. Fix a positive integer t. A partition of n is called a t−core partition of n if none of its hook numbers are multiples of t. Let ct(n) denote the number of t−core partitions of n. It has been conjectured that if t ≥ 4, then ct(n) > 0 for all n ≥ 0. In [7], the author proved the conjecture for t ≥ 4 even and for those t divisible by at least one of 5, 7, 9, or 11. Moreover if t ≥ 5 is odd, then it was shown that ct(n) > 0 for n sufficiently large. In this note we show that if k ≥ 2, then c3k(n) > 0 for all n using elementary arguments. A partition of a positive integer n is a nonincreasing sequence of positive integers with sum n. Here we define a special class of partitions. Definition 1. Let t ≥ 1 be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t−core partition of n. The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,4,5]. If t ≥ 1 and n ≥ 0, then we define ct(n) to be the number of partitions of n that are t−core partitions. The arithmetic of ct(n) is studied in [3,4]. The power series generating function for ct(n) is given by the infinite product: (1) ∞ ∑

Transitive permutation groups and irreducible linear groups

Abstract: IT was proved by M. F. Newman and the second author that there is a constant c such that each nilpotent transitive permutation group of degree d>2 can be generated by [cd/Vlogd] elements. Later J. D. Dixon and the second author showed that, for each field F which has finite degree over its prime subfield, there is a constant cr such that each finite nilpotent irreducible linear group of degree d over F can be generated by [cFd/Vlogd] elements. For a finite group G, let R(G) denote the product of the soluble radical and the generalized Fitting subgroup. Here we extend both results from nilpotent groups to the much wider class of all finite G such that R(G) = G. The proof easily reduces to dealing with groups that are either soluble or quasinilpotent. The soluble case involves a result of independent interest: there is a constant b such that, given a module of dimension a (over an arbitrary field) for a subgroup of index d in a finite soluble group, each submodule of the induced module can be generated by [abdlVlog d] elements. One of the steps towards the quasinilpotent case is the following theorem,' which we prove using the classification of finite simple groups. If G is any finite group and n is the composition length of the quotient of the generalized Fitting subgroup of G modulo the Frattini subgroup of R(C), then G can be generated by 5n elements.

Isometry Classes of Indecomposable Linear Codes

Abstract: In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian. We describe these classes as orbits and we demonstrate how they can be enumerated using cycle index polynomials. The necessary tools are already incorporated in SYMMETRICA, a (public domain) computer algebra package devoted to representation theory and combinatorics of symmetric groups and of related classes of groups. Moreover, we describe how systems of representatives of these classes can be evaluated using double coset methods.

Unbounded families and the cofinality of the infinite symmetric group

Abstract: In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.