Showing papers on "Symmetric group published in 1993"
TL;DR: The number of permutations with given cycle structure and descent set is shown to be equal to the scalar product of two special characters of the symmetric group.
353 citations
TL;DR: In this article, the second largest eigenvalues in terms of the eigen values of a comparison chain with known eigenvalue were derived for the symmetric random walk on a finite group.
Abstract: We develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution. The techniques gives bounds on the second largest (and other) eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues. The techniques yield sharp rates for a host of previously intractable problems on the symmetric group.
193 citations
TL;DR: One of the special cases of one of the stronger conjectures involving immanants develops from a generalization of the theory of permutations with restricted position which takes into account the cycle structure of the permutations.
166 citations
TL;DR: These results show that, in contrast to the case for (unsigned) permutations, these two statistics are not generally equidistributed.
Abstract: We derive multivariate generating functions that count signed permutations by various statistics, using the hyperoactahedral generalization of methods of Garsia and Gessel. We also derive the distributions over inverse descent classes of signed permutations for two of these statistics individually (the major index and inversion number). These results show that, in contrast to the case for (unsigned) permutations, these two statistics are not generally equidistributed. We also discuss applications to statistics on the wreath product Ck ≀ Sn of a cyclic group with the symmetric group.
130 citations
TL;DR: A general and closed-form relation is obtained here, in this equation the part involving RDM's has the same structure as that involving hole reduced density matrices.
Abstract: The commutation-anticommutation relations of q-electron operators imply a set of N representability conditions [A. J. Coleman, Rev. Mod. Phys. 31, 668 (1963)] for the corresponding q-order reduced density matrices (q-RDM) [C. Valdemoro, An. Fis. 79, 95 (1983); in Structure, Interaction and Reactivity, edited by S. Fraga (Elsevier, Amsterdam, 1992)]. From these conditions, a general and closed-form relation is obtained here. In this equation the part involving RDM's has the same structure as that involving hole reduced density matrices. This relation is the basis of a method for approximating a q-RDM in terms of the r-RDM's [C. Valdemoro, Phys. Rev. A 45, 4462 (1992)] with rq. The derivation of this relation can be simplified significantly by employing a graph method which is described here. These graphs are in a one-to-one correspondence with the elements of the symmetric group of permutations.
121 citations
01 Jan 1993
TL;DR: In this article, a family of representations unitaires du groupe symetrique denombrable S is described, i.e., a deformation de la representation reguliere T∞.
Abstract: Nous etudions une famille {T z :∈C} de representations unitaires du groupe symetrique denombrable S. C'est une deformation de la representation reguliere T∞. Les principaux resultats sont le calcul des caracteres des T z et, lorsque z∈Z, la decomposition de T z en une integrale directe de representations factorielles finies. En fait, au lieu des representations factorielles du groupe S, nous utilisns le langage equivalent des representations irreductibles de la paire de Gelfand (G, K)
117 citations
TL;DR: It is proved that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group S n, chosen randomly and independently, generate invariably S n with probability at least 1 − ∈.
Abstract: We prove that the probability i(n, k) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k, for k ≤ n/2. Using this fact, we prove that the fraction of elements of Sn belong to transitive subgroups other than Sn or An tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group Sn, chosen randomly and independently, generate invariably Sn with probability at least 1 − ∈. This confirms a conjecture of McKay.
86 citations
TL;DR: It is shown that, while a theorem of Rado implies a positive answer for the symmetric group, the general answer is negative, and the induced stratifications are nontrivial, and should be the subject of a future study.
60 citations
TL;DR: In this article, a number of linear congruences modulo r are proved for the number of partitions that are p-cores where p is prime, 5≤p≤23, and r is any prime divisor of 1/2(p-1).
Abstract: A number of linear congruences modulo r are proved for the number of partitions that are p-cores where p is prime, 5≤p≤23, and r is any prime divisor of 1/2(p-1). Analogous results are derived for the number of irreducible p-modular representations of the symmetric group S n . The congruences are proved using the theory of modular forms
01 Jan 1993
TL;DR: In this paper, a generalization of symmetric function theory in which we deal with trace functions on several matrices over ℂ is considered. But it is not shown how to obtain a linear character of the symmetric polynomials in n variables.
Abstract: The ring of symmetric polynomials in n variables may be interpreted as a ring of characters of the general linear group GL( n ) (see e.g. [4], §3·5). We consider here a generalization of symmetric function theory in which we deal with trace functions on several matrices over ℂ An important feature of the ‘single matrix variable’ (or classical) case is the characteristic isomorphism (see [5], chapter I, §7) ch: R = between the graded ring R , whose component R d in degree d is the character group of the symmetric group of degree d , and the ring of symmetric functions Λ in infinitely many variables (obtained from symmetric functions in n variables by taking the graded inverse limit with respect to n ).
TL;DR: In this paper, Cohen showed that 1 n is a lower bound for f(G) = |A|/|G|, where A is the set of elements of G that move every letter and G is a transitive permutation group of degree n. Lenstra's problem arose from his work on the number field sieve.
Abstract: In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1 n is such a bound. Lenstra’s problem arose from his work on the number field sieve [2]. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial over the integers, consider the proportion:
TL;DR: In this paper, the problem of finding all groups acting 2-arc transitively on finite connected graphs such that there exists a minimal normal subgroup that is nonabelian and regular on vertices is addressed.
Abstract: The paper addresses a part of the problem of classifying all 2-arc transitive graphs: namely, that of finding all groups acting 2-arc transitively on finite connected graphs such that there exists a minimal normal subgroup that is nonabelian and regular on vertices. A construction is given for such groups, together with the associated graphs, in terms of the following ingredients: a nonabelian simple group T, a permutation group P acting 2-transitively on a set Ω, and a map F : Ω→Tsuch that x e x−1 for all x ∈ F(Ω) and such that Tis generated by F(Ω). Conversely we show that all such groups and graphs arise in this way. Necessary and sufficient conditions are found for the construction to yield groups that are permutation equivalent in their action on the vertices of the associated graphs (which are consequently isomorphic). The different types of groups arising are discussed and various examples given.
TL;DR: In this article, it was shown that the automorphism group of generalized Reed-Muller codes is the general linear nonhomogeneous group (GLNG) and that the monomial group is the direct product of the GRLG with the multiplicative group of the alphabet field.
TL;DR: In this paper, a generalized permutation polytope is defined as the convex hull of the points in a polytopes whose coordinates are permutations of distinct numbers not necessarily distinct values.
Abstract: Exploratory graphical methods for fully and partially ranked data are proposed. In fully ranked data, $n$ items are ranked in order of preference by a group of judges. In partially ranked data, the judges do not completely specify their ranking of the $n$ items. The resulting set of frequencies is a function on the symmetric group of permutations if the data is fully ranked, and a function on a coset space of the symmetric group if the data is partially ranked. Because neither the symmetric group nor its coset spaces have a natural linear ordering, traditional graphical methods such as histograms and bar graphs are inappropriate for displaying fully or partially ranked data. For fully ranked data, frequencies can be plotted naturally on the vertices of a permutation polytope. A permutation polytope is the convex hull of the $n$! points in $\mathbb{R}^n$ whose coordinates are the permutations of $n$ distinct numbers. The metrics Spearman's $\rho$ and Kendall's $\tau$ are easily interpreted on permutation polytopes. For partially ranked data, the concept of a permutation polytope must be generalized to include permutations of nondistinct values. Thus, a generalized permutation polytope is defined as the convex hull of the points in $\mathbb{R}^n$ whose coordinates are permutations of $n$ not necessarily distinct values. The frequencies with which partial rankings are chosen can be plotted in a natural way on the vertices of a generalized permutation polytope. Generalized permutation polytopes induce a new extension of Kendall's $\tau$ for partially ranked data. Also, the fixed vector version of Spearman's $\rho$ for partially ranked data is easily interpreted on generalized permutation polytopes. The problem of visualizing data plotted on polytopes in $\mathbb{R}^n$ is addressed by developing the theory needed to define all the faces, especially the three and four dimensional faces, of any generalized permutation polytope. This requires writing a generalized permutation polytope as the intersection of a system of linear equations, and extending results for permutation polytopes to generalized permutation polytopes. The proposed graphical methods is illustrated on five different data sets.
TL;DR: In this article, four methods are described for enumerating digraphs with a given automorphism group: (1) a generating-function method based on subduced cycle indices, (2) a generator based on partial cycle indices (PCI), (3) an elementary superposition theorem and (4) a partial superposition method.
Abstract: Four methods are described for enumerating digraphs with a given automorphism group: (1) a generating-function method based on subduced cycle indices, (2) a generating-function method based on partial cycle indices, (3) a method based on the elementary superposition theorem, and (4) a method based on the partial superposition theorem. All of these methods are based on the concept of unit subduced cycle indices and construct a set of versatile tools for combinatorial enumeration. They are applied to the enumeration of five-vertex digraphs with a given automorphism group. The table of marks and its inverse for the symmetric group of degree 5 are recalled. The table of USCIs of this roup is obtained.
TL;DR: In this paper, independent, H distributed families of random unitaries in symmetric groups are shown to be asymptotically free, where H is the number of units in the group.
Abstract: We prove that independent, H distributed families of random unitaries in symmetric groups are asymptotically free
01 Sep 1993
TL;DR: In this paper, the authors present a system for computing with graphs and groups by L. H. Babai, E. M. Luks, and A. W. York.
Abstract: Computing composition series in primitive groups by L. Babai, E. M. Luks, and A. Seress Computing blocks of imprimitivity for small-base groups in nearly linear time by R. Beals Fast Fourier transforms for symmetric groups by M. Clausen and U. Baum From hyperbolic reflections to finite groups by J. H. Conway Combinatorial tools for computational group theory by G. Cooperman and L. Finkelstein Efficient computation of isotypic projections for the symmetric group by P. Diaconis and D. Rockmore Constructing representations of finite groups by J. D. Dixon A graphics system for displaying finite quotients of finitely presented groups by D. F. Holt and S. Rees Random remarks on permutation group algorithms by W. M. Kantor Application of group theory to combinatorial searches by C. W. H. Lam Permutation groups and polynomial-time computation by E. M. Luks Parallel computation of Sylow subgroups in solvable groups by P. D. Mark Computation with matrix groups over finite fields by C. E. Praeger Asymptotic results for permutation groups by L. Pyber Computations in associative algebras by L. Ronyai Cayley graphs and direct-product graphs by A. L. Rosenberg Group membership for groups with primitive orbits by N. Sarawagi, G. Cooperman, and L. Finkelstein PERM: A program computing strong generating sets by A. Seress and I. Weisz Complexity issues in infinite group theory by C. C. Sims GRAPE: A system for computing with graphs and groups by L. H. Soicher Implications of parallel architectures for permutation group computations by B. W. York.
TL;DR: In this article, an algorithm for the construction of a complete system of representatives of t-designs with given parameters t − (v, k, λ) and prescribed full automorphism group A is presented.
Abstract: We introduce an algorithm for the construction of a complete system of representatives of t-designs with given parameters t − (v, k, λ) and prescribed full automorphism group A It is based on the following observation published by Kramer and Mesner in 1976: The t − (v, k, λ) designs admitting automorphism group A are exactly the 0-1-solutions of the following system of linear equations
M are incidence matrices, which we compute by means of double cosets Representing the set of all solutions of the above system of equations implicitly by a graph gives us the possibility either to extract the solutions explicitly or to compute their precise numbers, which often are very big We use the lattice of overgroups of A in the full symmetric group Sv for the construction or enumeration of the isomorphism types of the t-designs with full automorphism group A from these solutions To the best of our knowledge our approach for the first time allows one to compute the precise number of isomorphism types or even these designs themselves for substantial numbers We determined the (number of) isomorphism types for many known parameter sets and found new simple 6-designs with parameters
and full automorphism group PΓL2(27) We constructed all isomorphism types of these designs; their precise numbers are 3,367,21 743,38 277, respectively © 1993 John Wiley & Sons, Inc
TL;DR: In this article, the reduced notation for irreducible representations of the symmetric group S, interpreted in terms of symmetric formal series and vertex operators, is used to prove a number of properties of reduced Kronecker products and inner plethysms in an n-independent manner.
Abstract: The reduced notation for irreducible representations of the symmetric group S, is interpreted in terms of symmetric formal series and vertex operators, and is used to prove a number of properties of reduced Kronecker products and inner plethysms in an n-independent manner. Conditions for self-associativity of Kronecker products and inner plethysms are established. Reduced inner plethysms are developed and applied to the question of non-simple phase groups among the symmetric Sn and alternating An groups.
TL;DR: In this paper, it was shown that all finite groups with a unique element of order two such that the order of the group is not divisible by three are solvable and thus, except for the quaternion group, have symmetric sequencings.
Abstract: All finite solvable groups that have symmetric sequencings are characterized. Let G be a finite solvable group. It is shown that G has a symmetric sequencing if and only if G has a unique element of order two and is not the quaternion group. All finite groups with a unique element of order two such that the order of the group is not divisible by three are solvable and thus, except for the quaternion group, have symmetric sequencings. A crucial step used in the proof of these facts is a construction showing that if a finite group H has a normal subgroup C of odd order such that H/C admits a 2-sequencing, then H admits a 2-sequencing. The results of this article can be viewed as generalizing a theorem of Gordon about Abelian groups and as extending the idea of a starter, suitably modified, to a large class of groups of even order by showing the existence of the required object. © 1993 John Wiley & Sons, Inc.
01 Mar 1993
TL;DR: In this article, an equivariant collection of contractions of the compactified Culler-Vogtmann outer space X was constructed, and it was shown that any finite subgroup of the outer automorphism group of a free group fixes a contractible subset of X.
Abstract: We construct an equivariant collection of contractions of the compactified Culler-Vogtmann outer space X„ . As a consequence, we prove that any finite subgroup of the outer automorphism group of a free group fixes a contractible subset of X„ .
TL;DR: In this paper, it was shown that for every n ≥ 3, there is an n long sequence of adjacent transpositions, each except the first adjacent to its predecessor, and such that πn equal to the identity.
Abstract: Let x 1 x 2…xi xi+1 …xn be a permutation of {1,2,…,n} written in one line notation. An "i-adjacent interchange" applied to this permutation produces the permutation x 1 x 2…xi+1 xi …xn obtained from the first permutation by interchanging the symbols in positions i and i+1. We denote such an interchange by (i,i+1), where i=1,…,n−1. The permutation x 1 x 2…xi+1 xi …xn results from x 1 x 2…xi xi+1 …xn by right multiplication by the transposition (i,i+1) in the symmetric group Sn . Two adjacent transpositions (i,i+1) and (j,j+1) are themselves adjacent if either i=j+1 or j=i+1. In this paper, we show that that for every n≥3, there is an n long sequence (f(1),f(1)+1),…,(f(n!),f(n!)+1) of adjacent transpositions, each except the first adjacent to its predecessor, the first adjacent to the last, and such that the set of products of transpositions {πt:πt=(f(1),f(1)+1)…(f(t),f(t)+1)t=1,…,n!} is Sn , with the final product πn equal to the identity. We call such a sequence of transpositions a doubly adjacent Gray co...
TL;DR: In this paper, the mathematical formulation of resonance theory is discussed and a new basis for an irreducible representation of the unitary group, called bonded tableau, can be constructed to describe a resonance structure correspondingly.
Abstract: In the present paper, the mathematical formulation of resonance theory is discussed. In fact, resonance theory is a representative form of the unitary group approach. Using the standard projection operator of the symmetric group, a new basis for an irreducible representation of the unitary group, called bonded tableau, can be constructed to describe a resonance structure correspondingly. The relationships between bonded tableau and Weyl tableaux and between valence bond and molecular orbital approaches are revealed. Finally, test calculations on ozone and benzene are performed.
TL;DR: In this paper, the authors evaluate the generating function of the symmetric group with respect to five statistics and derive the continued fraction expansion of its ordinary generating function by the methods of Stieltjes and Rogers, which implies especially the combinatorial interpretation of the moments of the orthogonal Sheffer polynomials.
Abstract: In this paper, we evaluate the generating function of the symmetric group with respect to five statistics. The continued fraction expansion of its ordinary generating function is then derived by the methods of Stieltjes and Rogers, which implies especially the combinatorial interpretation of the moments of the orthogonal Sheffer polynomials.
TL;DR: In this article, it was shown that the Lie group of type E 8 hw is a unique conjugacy class of subgroups isomorphic to the finite simple group L(2 61) of linear fractional transformations over a field of 6 1 elements.
Abstract: In this paper, we prove t h a t the Lie group of type E 8 hw a unique conjugacy class of subgroups isomorphic to the finite simple group L(2 61) of linear fractional transformations over t h e field of 6 1 elements. This result settles the last open case of a conjecture made by B. Kostant in 1983 [Iiost] concerning the occurrence of the finite simple group in a complex simple Lie group with Coxeter number h. It also settles a case left open in the classifications of finite simple subgroups of obtained by the first two authors in 1987.