Showing papers on "Symmetric group published in 1987"
TL;DR: In this article, the maximal subgroups of alternating groups A and S are known for several classes of degrees n, where S is an alternating group A, and X is a simple group A.
295 citations
TL;DR: A substantial amount of work has been devoted to finding the maximal subgroups of finite simple groups and their automorphism groups as discussed by the authors, and the main result of this nature appears in Wiman [42] and Moore [33].
177 citations
TL;DR: In this paper, the character theory of the symmetric group is used to derive properties of the number of permutations, with k cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class.
Abstract: The character theory of the symmetric group is used to derive properties of the number of permutations, with k cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class. For the conjugacy class of fixed point free involutions, this problem has application to the analysis of singularities in surfaces. 1. Introduction. For nonnegative integers k and N, and a partition \p of N, let ef denote the number of permutations w on N symbols such that it has exactly k cycles, and such that it can be expressed as a product of an arbitrary, but fixed, cycle of length N and a permutation in the conjugacy class indexed by \p. The purpose of this paper is to derive the generating function for these numbers, and to obtain some of their properties. The method makes direct use of combinatorial and algebraic properties of the group algebra of the symmetric group. A special case of this problem is of particular interest. Let e(p)(n) denote the number e\ when \p indexes the conjugacy class of permutations on pn symbols, with n cycles of length p. The matter of calculating ek2
) arose in connection with work by Harris and Morrison (4) on singularities in surfaces. It has also occurred indirectly in the work of Gross (2) on graph embeddings. Harer and Zagier (3) have shown, by an independent method, that the sequence e(k2)(n) for k, n > 1 satisfies a three-term linear recurrence equation with coefficients which are polynomials in n. To fix ideas, note that for n = 2 the permissible permutations are
123 citations
01 May 1987
TL;DR: It is known that the cyclic permutations generate Sn, and it easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n ( = Singn) is equal to the rank of Sn as discussed by the authors.
Abstract: It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutationsgenerate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n ( = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.
105 citations
TL;DR: A generalization of the usual column-strict tableaux is presented as a natural combinatorial tool for the study of finite dimensional representations of GL n and these objects are called rational tableaux since they play the same role for rational representations ofGL n as ordinary tableaux do for polynomial representations.
98 citations
01 Jan 1987
TL;DR: Ozaydin et al. as discussed by the authors proposed an equivariant map for the symmetric group, which is based on the ODE algorithm for symmetric groups, and proved its correctness.
Abstract: M. Ozaydin, Equivariant maps for the symmetric group, unpublished preprint, University of Wisconsin-Madison, 1987, 17
84 citations
Book•
01 Jan 1987
TL;DR: In this article, the authors present a survey of the conventional methods of symmetry adaptation in quantum chemistry, including the Symmetric Group, the Unitary Group, and the Freeon Many-Body Theory.
Abstract: 1. Survey of the Conventional Methods. 2. The Symmetric Group. 3. Applications of the Symmetric Group in Quantum chemistry. 4. The Unitary Group. 5. The Unitary Group Formulation of Quantum Chemistry. 6. Computational Applications. 7. Theoretical Organic Chemistry (A Pedagogical Chapter). 8. Symmetry-Adaptation. 9. Freeon Many-Body Theory. 10. Fermion Many-Body Theory. Bibliography. Author Index. Subject Index.
72 citations
TL;DR: In this paper, the maximum order of an element of a finite symmetric group is analyzed in terms of the number of elements in the group and the order in which the elements are ordered.
Abstract: (1987). The Maximum Order of an Element of a Finite Symmetric Group. The American Mathematical Monthly: Vol. 94, No. 6, pp. 497-506.
65 citations
Book•
30 Jan 1987
TL;DR: In this article, the authors consider linear groups over finite fields and show that linear groups with p-rank less than or equal to 2 can be represented by vector bundles over the classifying space.
Abstract: 1. Group cohomology 2. Products and change of group 3. Relations with subgroups and duality 4. Spectral sequences 5. Representations and vector bundles 6. Bundles over the classifying space for a discrete group 7. The symmetric group 8. Finite groups with p-rank less than or equal to 2 9. Linear groups over finite fields.
60 citations
TL;DR: In this article, it was shown that the monodromy action on the set of 97 on a fiber is irreducible, and that the family of all cffs on all fibers of a fiber of genus g on a moduli space B of the modulus space My is a finite cover of B. In particular, the branch locus of this cover plays a fundamental role in the even-genus case in the case r = 1.
Abstract: — Let g, r, and d be positive integers such that g=(r-{-1) (g—d-\-r), so that the general curve of genus g has only finitely many g^s. We will show in this paper that for suitable families of curves ^ -> B, the family of all cffs on all fibers of ^ -> B is irreducible. We do this by analyzing the monodromy action on the set of 97 on a fibre, using a degeneration to reducible curves and our technique of limit series [198?^]. In the case r = 1 we prove the sharper statement that the monodromy is the full symmetric group, a result motivated by a problem posed by Verdier, and applied by him in the study of harmonic maps from 2 to S (Verdier [198?]). If we take ^ to be the universal curve over a suitable open set B of the moduli space My then the family of c^'s is a finite cover of B, and the branch locus of this cover (in the case r=l), analyzed through the tools developed in this paper, plays a fundamental role in the even-genus case in our proof [198?^] that Jig has general type for all g 24.
58 citations
TL;DR: Given generators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time.
Abstract: Given generators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.
TL;DR: In this paper, the exceptional simple groups of Lie type and their automorphism groups were studied and the maximal factorizations of these groups were shown to be non-abelian.
TL;DR: In this article, it was shown that Thomae's identity between two 3F2 hypergeometric series of unit argument together with the trivial invariance under separate permutations of numerator and denominator parameters implies that the symmetric group S5 is an invariance group of this series.
Abstract: It is shown that Thomae’s identity between two 3F2 hypergeometric series of unit argument together with the trivial invariance under separate permutations of numerator and denominator parameters implies that the symmetric group S5 is an invariance group of this series. A similar result is proved for the terminating Saalschutzian 4F3 series, where S6 is shown to be the invariance group of this series (or S5 if one parameter is eliminated by using the Saalschutz condition). Here Bailey’s identity is realized as a permutation of appropriately defined parameters. Finally, the set of three‐term relations between 3F2 series of unit argument discovered by Thomae [J. Thomae, J. Reine Angew. Math. 87, 26 (1879)] and systematized by Whipple [F. J. Whipple, Proc. London Math. Soc. 23, 104 (1925)] is shown to be transformed into itself under the action of the group S6×Λ, where Λ is a two‐element group. The 12 left cosets of S6×Λ with respect to the invariance group S5 are the structural elements underlying the three‐...
01 Jan 1987
TL;DR: In this paper, the authors gave conditions under which the Galois group of the polynomial Xn + aX1 + b over the rational number field Q is isomorphic to the symmetric group Sn of degree n.
TL;DR: In this article, a general recursion is given for computing the irreducible decomposition of certain characters of SL( n, C) from a combinatorial point of view.
Abstract: Some problems concerning the decomposition of certain characters of SL( n, C) are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon's so-called "Macdonald complex." A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of n. In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter) q-Dyson Theorem.
TL;DR: A short proof of the fact, known to Jordan, that any permutation group of degree n other than the symmetric and alternating groups is at most c(log n)2(log log n)-fold transitive.
TL;DR: In this paper, a method for obtaining Formal Group Laws from the structure constants of Affine Kac-Moody groups and then applying a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group.
Abstract: We describe a method for obtaining Formal Group Laws from the structure constants of Affine Kac-Moody groups and then apply a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group. As an intermediate step we get an explicit expression for two-cocycles on Loop Groups. The programme is applied to the AffineSU(2) group.
TL;DR: In this paper, a complete description of trace identities for matrix superalgebras where and are square matrices of orders and respectively over the even elements of a Grassmann algebra with countably many generators is given.
Abstract: A complete description is given of trace identities for matrix superalgebras where and are square matrices of orders and respectively over the even elements of a Grassmann algebra with countably many generators, while and are and rectangular matrices respectively over the odd elements of . A relation is found between multilinear trace identities of degree in the algebra and irreducible representations of a symmetric group of order . It is proved that over a field of characteristic zero all trace identities of follow from identities of degree that hold in that algebra. For every algebra over a field of arbitrary characteristic a central polynomial is given explicitly.Bibliography: 7 titles.
TL;DR: The determination of polynomials over @?(t) with a given primitive nonsolvable permutation group of degree d =< 15 as Galois group is completed.
TL;DR: In this article, it was shown that the monodromy of the family of Riemann surfaces acts as the full symmetric group on the Weierstrass points of a general curve.
Abstract: We show that the monodromy of the family of curves (Riemann surfaces) acts as the full symmetric group on the Weierstrass points of a general curve. The proof uses a degeneration to certain reducible curves, and the theory of limit series developed in our (1986, 1987a, b). Some of the monodromy is actually constructed by fixing a (reducible) curve and varying its “canonical” series.
TL;DR: In this article, a semigroup S of total or partial transformations of a set X is called -normal if hSh-1 = S, for all h in, the symmetric group on X.
Abstract: We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh-1 = S, for all h in , the symmetric group on X. For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.
01 Jan 1987
TL;DR: In this paper, it was shown that every finite semigroup whose idempotents commute divides a finite inverse semigroup (Ash's theorem), and, more generally, that any semigroup with idempots form a subsemigroup, which is a semigroup that is a sub-semigroup of a finite orthodox semigroup.
Abstract: We give a new proof that every finite semigroup whose idempotents commute divides a finite inverse semigroup (Ash’s theorem), and, more generally, we prove that every finite semigroup whose idempotents form a subsemigroup divides a finite orthodox semigroup.
01 Jan 1987
TL;DR: In this article, a semigroup S of total or partial transformations of a set X is called -normal if hSh-1 = S, for all h in, the symmetric group on X.
Abstract: We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh-1 = S, for all h in , the symmetric group on X. For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.
TL;DR: In this paper, a symmetric group approach is presented for performing large scale, direct configuration interaction (CI) studies of electronic correlation effects in molecules, and two alternative graphical methods are described for representing very large sets of lexically ordered orbital configurations.
Abstract: A symmetric group approach is presented for performing large scale, direct configuration interaction (CI) studies of electronic correlation effects in molecules. Two alternative graphical methods are described for representing very large sets of lexically ordered orbital configurations. One is suited for truncations in the number of core orbitals and the other for changes in the number of virtual orbitals. The Hamiltonian matrix elements are constructed using a graphical approach for localizing the one-and two-electron contributions. Compact and direct algorithms are given for calculating the irreducible representation matrices of S(N) for the “line-up” permutations. The program implementation of the CI procedure, PEDICI, calculates directly firstorder molecular one-electron properties and transition probabilities from a multireference space for general excitation schemes.