Showing papers on "Symmetric group published in 1977"
Book•
28 Oct 1977
TL;DR: In this article, the authors present an introductory text to the theory of group characters, where the structure of an abstract system is elucidated by a unique set of numbers inherent in the system.
Abstract: To an algebraist the theory of group characters presents one of those fascinating situations, where the structure of an abstract system is elucidated by a unique set of numbers inherent in the system. But the subject also has a practical aspect, since group characters have gained importance in several branches of science, in which considerations of symmetry play a decisive part. This is an introductory text, suitable for final-year undergraduates or postgraduate students. The only prerequisites are a standard knowledge of linear algebra and a modest acquaintance with group theory. Especial care has been taken to explain how group characters are computed. The character tables of most of the familiar accessible groups are either constructed in the text or included amongst the exercise, all of which are supplied with solutions. The chapter on permutation groups contains a detailed account of the characters of the symmetric group based on the generating function of Frobenius and on the Schur functions. The exposition has been made self-sufficient by the inclusion of auxiliary material on skew-symmetric polynomials, determinants and symmetric functions.
179 citations
101 citations
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TL;DR: In this article, the outer automorphism group of a free group is studied in the case that the group G and F are finite and F is finitely generated, respectively, and one obtains a finite permutation representation of Out (F).
Abstract: Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free group.
67 citations
54 citations
TL;DR: In this paper, the subgroups that arise in phase transitions from a high-symmetry phase are characterized as those subgroup that are maximal with respect to the property of acting trivially on a given non-zero subspace U of the representation space Mi of a given irreducible representation Ti of H. The use of permutation representations considerably simplifies the theory.
Abstract: The subgroups that arise in phase transitions from a high-symmetry phase are characterized as those subgroups that are maximal with respect to the property of acting trivially on a given non-zero subspace Ui of the representation space Mi of a given irreducible representation Ti of H. In the case of subgroups of finite index the problem is reduced to that of studying faithful irreducible representations of finite groups. The use of permutation representations considerably simplifies the theory. Tables of the equitranslation epikernels of the space groups are given.
53 citations
01 Jan 1977
TL;DR: In this paper, the authors present a technique that enables a computer to process identities in non-associative algebras, where identities are expressed as elements of the group ring on the symmetric group and then transferred to matrices by the well-known isomorphisms of group rings.
Abstract: Publisher Summary This chapter presents a technique that enables a computer to process identities. The examples used are from nonassociative algebras, but the method is general and can be applied to many other situations. Identities are expressed as elements of the group ring on the symmetric group and then are transferred to matrices by the well-known isomorphisms of group rings. This approach has several advantages over the standard approach, which uses a system of linear equations. The system indicates whether a desirable identity is a consequence of other assumed identities. If the identity is not a consequence of the assumed identities, it tells how close it is to being true. This lets the investigator know the routes that are worth pursuing. Second, it uses very little array area. Identities involving four unknowns are processed, with the largest array being a 3 × 3 matrix.
33 citations
TL;DR: In this paper, the structure of symmetric balanced incomplete block designs (SBIBDs) is studied and applications to near planes (Fast-blockplane) and Baer sub-designs of SBIBD are given.
28 citations
21 citations
TL;DR: The object is to enumerate graphs in which the points or lines or both are assigned positive or negative signs and the solutions to all of these counting problems can be expressed as special cases of one general formula involving the concatenation of the cycle index of the symmetric group with that of its pair group.
Abstract: Our object is to enumerate graphs in which the points or lines or both are assigned positive or negative signs. We also treat several associated problems for which these configurations are self-dual with respect to sign change. We find that the solutions to all of these counting problems can be expressed as special cases of one general formula involving the concatenation of the cycle index of the symmetric group with that of its pair group. This counting technique is based on Polya's Enumeration Theorem and the Power Group Enumeration Theorem. Using a suitable computer program, we list the number of graphs of each type considered up to twelve points. Sharp asymptotic estimates are also obtained.
TL;DR: It is shown that H is normal in G = Γ(T)v for any tree T and any vertex v, if and only if, for all vertices u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N.
TL;DR: In this paper, the authors adopt the notation of [3] so that when h = (A, A,,..., Ak) is a proper partition of n, MA is the permutation module of the symmetric group 6, on the subgroup GA1 x GA2 x 1.. x GA1 over some field F. The Specht module, 9, is a submodule of MA.
TL;DR: In this paper, the transformation from the geneological spin basis to the Jahn-Serber basis is considered as an illustrative example, and transformation matrices for the representations of the spin permutation group may be easily derived using graphical methods of angular momentum theory.
TL;DR: The structure of the group algebra KG of an arbitrary periodic group G such that the multiplicative group U(KG) is solvable is completely determined in this paper, and the structure of KG is known.
Abstract: The structure of the group algebra KG of an arbitrary periodic group G such that the multiplicative group U(KG) is solvable is completely determined.
TL;DR: Scholz as mentioned in this paper considered the problem of algebraic number fields whose Galois group G is a Frobenius group and showed that the structure of the unit group and ideal class group of K is determined by that; of the subfields fixed by N and by a complement F, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when g is a dihedral group of order 2n for an odd integer n.
Abstract: Let K/k be a normal extension of algebraic number fields whose Galois group G is a Frobenius group. Then K/k is said to be a Frobenius extension. Most of the structure of the unit group and of the ideal class group of K is determined by that; of the subfields fixed by the Frobenius kernel N and by a complement F. Here this is investigated when G is a maximal or metacyclic Frobenius group. In particular, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when G is a dihedral group of order 2n for an odd integer n. A. Scholz, taking n = p = 3, was the first to consider this problem.
TL;DR: In this article, the authors describe a scenario where a group of people are living in a country where they are confronted with the problem of "missing links" and "missing connections".
Abstract: 本文中分析了态空间置换群g f 表示论中的矛盾。指出了原因在于当态指标有重复时,g f 的单个群元算符没有确定作用。根据文献[1]给出了这种情形下g f 的亚标准基的定义、标志和求法,并证明了g f 亚标准基同时就是酉群的Gelfand基。
TL;DR: In this article, the existence of a subgroup S ∼-S 10, the symmetric group of degree 10, was proved and a complete transform table for the sporadic simple group F 22 was given.
01 Jan 1977
TL;DR: In this article, the authors present the definition and algebraic properties of certain square symbols appearing in the finite irreducible representations of the group U(n) and its representations.
Abstract: Publisher Summary This chapter presents the definition and the algebraic properties of certain square symbols appearing in the finite irreducible representations of the group U(n). This concept is linked to properties of the symmetric group S(N) and its representations. The recent developments of the technique of S(N) for many-body physics are closely related to the properties of these square symbols. The chapter discusses high-dimensional IR of S(N) for which new techniques are required. The concept of double cosets is very important as it is directly linked to the exchange properties. The properties known for S(N) can serve to bring out new features of the IR of U(n), such as Regge symmetry.
TL;DR: In this paper, the authors describe all compact symmetric subgroups of the orthogonal group O n which contain the permutation groupS n, and each one can be realized as the group of all isometries on somen-dimensional Banach space.
Abstract: We describe all compact symmetric subgroups of the orthogonal groupO
n which contain the permutation groupS
n. There are seven such groups, each one can be realised as the group of all isometries on somen-dimensional Banach space.
TL;DR: In this paper, a method for the determination of the representation matrices of the spin permutation group (symmetric group) is presented, where the representation matrix is derived from the permutation matrix.
Abstract: A method for the determination of the representation matrices of the spin permutation group (symmetric group) is presented.
TL;DR: In this article, the relations between the Waller-Hartree spin-free method and the symmetric group theory are given, and it is shown that the Gallup method is a special case of ours with S = M.
Abstract: The relations between the Waller–Hartree spin-free method and the symmetric group theory are given. It is shown that the Gallup method is a special case of ours with S = M. Furthermore, all the irreducible representation matrices and other matrices needed are written explicitly in terms of Sanibel coefficients which makes the method more useful. However, it was shown that the cases with S ≠ M for the spin-free pure spin states might be beyond the power of the symmetric group theory.
TL;DR: In this paper, the authors present identities on generating functions for multisectioned partitions of integers by developing in the language of partitions some powerful and essentially combinatorial techniques from the literature of principal differential ideals.
Abstract: This paper presents identities on generating functions for multisectioned partitions of integers by developing in the language of partitions some powerful and essentially combinatorial techniques from the literature of principal differential ideals. D. Mead has stated in Vol. 42 of this journal that one can obtain interesting combinatorial relations by constructing different vector space bases for a subspace of a differential ring and using the fact that the cardinality of all bases is the same. The results of the present paper are of this nature. In particular, we enumerate certain sets of ordered pairs of generalized tableaux that have a central role in Mead's paper. Tableaux were used by A. Young and others to study the structure of the symmetric groups Sn. In [3], D. Knuth used an "insertion into tableau" construction of C. Schensted to give a direct 1-to-l correspondence between "generalized permutations" and ordered pairs of "generalized Young tableaux" having the same shape. In [5], Mead independently proved the existence of such a bisection while developing a new vector space basis for the ring of differential polynomials in n independent differential indeterminates. Mead's paper deals with principal differential ideals generated by Wronskians and used determinantal identities going back to Gayley. The ordered pairs of generalized tableaux used by Mead appear in a more general setting in the paper [1] by Doubilet, Rota, and Stein.
TL;DR: In this paper, a doubly transitive permutation group of degree p 2 + 1, p a prime, is proved to be doubly primitive for p ≠ 2, and if such a group is not triply transitive then either it is a normal extension of P S L (2, p 2 ) or the stabilizer of a point is a rank 3 group.
Abstract: A doubly transitive permutation group of degree p 2 + 1, p a prime, is proved to be doubly primitive for p ≠ 2. We also show that if such a group is not triply transitive then either it is a normal extension of P S L (2, p 2 ) or the stabilizer of a point is a rank 3 group.
01 Apr 1977
TL;DR: It is proved that if n ⩾ 3, then the symmetric group Sn is not Z-sequenceable.