Showing papers on "Symmetric group published in 1986"
114 citations
TL;DR: In this article, a generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions, and applications are given to Ulam's problem on longest increasing subsequences and to a law of large numbers for representations.
Abstract: Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and central indecomposable measures are investigated. A generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions. Applications are given to Ulam’s problem on longest increasing subsequences and to a law of large numbers for representations. An analogous theory for other graphs is discussed.
89 citations
TL;DR: An axiomatic definition of a quantum monodromy matrix and the representations of its corresponding Hopf algebra are discussed in this article, where the connection between the quantum inverse transform method and the representation theory of a symmetric group is considered.
Abstract: An axiomatic definition of a quantum monodromy matrix and the representations of its corresponding Hopf algebra are discussed. The connection between the quantum inverse transform method and the representation theory of a symmetric group is considered. A new approach to the completeness problem of Bethe vectors is also given.
85 citations
TL;DR: Throughout this paper Q will denote an infinite set, S:= Sym(Q) and G is a subgroup of S, and working in ZFC, set theory with Axiom of Choice (AC), the subgroups G with S : G < 2 are sought.
Abstract: Throughout this paper Q will denote an infinite set, S:= Sym(Q) and G is a subgroup of S. If n: = |Q|, the cardinal of Q, then |.S| = 2\". Working in ZFC, set theory with Axiom of Choice (AC), we shall be seeking the subgroups G with \\S: G\\ < 2\". If A £ Q then S^ (respectively G{A}) denotes the setwise stabiliser of A in S (respectively in G); 5(A) and G(A) denote pointwise stabilisers; we identify S(A) with Sym(fi —A). A subset £ of Q such that |Z| = |Q — 1 | = |O| is known as a moiety ofQ. Suppose now that |Q| = n = Ko. If there is a finite subset A of Q such that 5(A) ^ G then certainly \\S:G\\ ^ Xo. Our theme is a rather strong converse:
77 citations
TL;DR: Etude des partitions en blocs de taille paire et impaire and des caracteres insolites des groupes symetriques as mentioned in this paper. But this is not the case in the present paper.
Abstract: Etude des partitions en blocs de taille paire et impaire et des caracteres insolites des groupes symetriques. Demonstration d'une conjecture de R.P. Stanley
71 citations
TL;DR: In this paper, the authors introduce the notion of primary subgroups and local subgroups, and discuss the relation between them and the centralizers of elements of elements in a given subgroup.
Abstract: CONTENTS Introduction § 1. Primary subgroups § 2. Local subgroups § 3. Conjugacy classes and centralizers of elements § 4. Embeddings § 5. Intermediate subgroups § 6. Permutation representations § 7. Maximal subgroups § 8. Other subgroups References
65 citations
TL;DR: In this article, the outer automorphism groups of both the mapping class groups and extended mapping classes of closed, connected, orientable surfaces of negative Euler characteristic are described.
Abstract: Ivanov of the Leningrad Branch of the Steklov Mathematical Institute has shown that the outer automorphism groups of surface mapping class groups are finite. In this report, we shall give explicit descriptions of the outer automorphism groups of both the mapping class groups and extended mapping class groups of closed, connected, orientable surfaces of negative Euler characteristic. (The extended mapping class groups are the extensions of the mapping class groups by the orientation reversing isotopy classes.) For surfaces of genus greater than two, these are, respectively, a cyclic group of order two and the trivial group. For a surface of genus two, these are both noncyclic groups of order four. (Once again, the hypergeometric involution in genus two plays a unique role.)
65 citations
TL;DR: It is proved that for n≥2, the length of every subgroup chain in Sn is at most 2n-3, which has applications to worst case complexity estimates for permutation group algorithms.
Abstract: We prove that for n≥2, the length of every subgroup chain in Sn is at most 2n-3. The proof rests on an upper bound for the order of primitive permutation groups, due to Praeger and Saxl. The result has applications to worst case complexity estimates for permutation group algorithms.
64 citations
01 Jan 1986
48 citations
01 Jan 1986
TL;DR: In this paper, a conjecture concerning the p-block structure of the projective representations of the symmetric group Sn was formulated in terms of these concepts which corresponds to the well-known, but long proved, Nakayama 'conjecture' for the linear representations of Sn.
Abstract: In [6] the first author introduced some combinatorial concepts involving Young diagrams corresponding to partitions with distinct parts and applied them to the projective representations of the symmetric group Sn. A conjecture concerning the p-block structure of the projective representations of Sn was formulated in terms of these concepts which corresponds to the well-known, but long proved, Nakayama ‘conjecture’ for the p-block structure of the linear representations of Sn. This conjecture has recently been proved by Humphreys [1].
35 citations
TL;DR: In this paper, the authors studied three finite quotients of the sequence of braid groups {B n;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐B n pass to well-defined equivalence classes in the quotient.
Abstract: This paper studies three finite quotients of the sequence of braid groups {B
n;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐B
n pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial on the corresponding classes.
01 Feb 1986
TL;DR: In this article, it was shown that a bar construction on a commutative d. algebra over a field I of characteristic zero induces a d. Hopf algebra isomorphism.
Abstract: An explicit formula for a canonical splitting s: Q4(9 ) ?4(& ) of the projection 4(l ) Q4(l ) of the bar construction on a commutative d.g. algebra 4' onto its indecomposables is given. We prove that s induces a d.g. algebra isomorphism A(Q4(9 )) R(o ) and that H(QtI(4`)) is isomorphic with QH(-4( )) If 6& is an augmented commutative d.g. algebra over a field I of characteristic zero, then the bar construction 4(9 ) on 4is a commutative d.g. Hopf algebra. Denote the augmentation ideal of 4(9 ) by If4( 61) (or I). The indecomposable elements of 4(6 ) are defined to be Q_(0 ) = I/I2. Since 2(9 ) = I D I?f(& ), there is a natural projection 4( 4') -Q4(& ). Our main result is the following. THEOREM. If & is a commutative d. g. algebra over the field I of characteristic zero, then there is a natural splitting s: Q2(9 ) -3 It4(9 ) of the natural projection rr: Ig(& ) -Q?6(9 ). The splitting s commutes with the differentials. Moreover, the map A(Q-4(&`)) --1(9 ) induced by s, from the free commutative d.g. algebra generated by Q2(6 ) into .4(& ), is a d.g. algebra isomorphism. In fact, the idempotent y = s o 7T is given by the formula n -y[all l an] = Y. E Y, (_l) m1 e( ) [a,(,)l ... la,(n)], m= 1 r1 +?r. = n aosh(r1,.* , rm) where sh(r1, .. ., rm) denotes the shuffles of (1, .. ., m } of type (r1, . . ., rm) and where E: -M { -1, 1 is the representation of the symmetric group obtained by assigning weight -1 + deg a1 to a1. COROLLARY. The natural map QH (?4(` )) -H (Q?1(9 )) is an isomorphism. 0 One version of the Poincare-Birkhoff-Witt (P.B.W.) theorem (cf. [6, appendix B]) states that if A is a commutative d.g. Hopf algebra, then there is a natural d.g. coalgebra isomorphism S(PA) A between A and the symmetric coalgebra on the primitives PA of A. Thus, the assertion that there is a d.g. algebra isomorphism Received by the editors March 20, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P62.
TL;DR: All distance-transitive representations of the symmetric groups Sn are classified and all pairs (G, Γ) such that G acts distance- transitively on Γ and G = Sn for some n are described.
TL;DR: Using the theory of Burnside rings, various canonical families of congruence relations hold for characters of the symmetric group and characterize the ring generated by them as a subring of the ring of integer valued class functions as discussed by the authors.
TL;DR: A complete group presentation consists of a set of generators and a setof replacement rules generating a well-founded and confluent relation on words, thereby solving the word problem for this presentation.
TL;DR: It is proved the every finite group is isomorphic to the full automorphism group of some perfect binary single error correcting code.
TL;DR: In this article, a model of quantum nonlinear wave interaction is solved algebraically, where the eigenstates are expressed in terms of symmetric group characters as quantum realizations of Schur functions.
TL;DR: In this paper, the authors give a different view of a theorem of Solomon [3] on Coxeter groups (finite groups of symmetries of R generated by reflections) and give a shorter and apparently more elementary proof which does not reduce to Etienne's proof in the special case of the symmetric group.
Abstract: The main purpose of this note is to give another proof and a different view of a theorem of Solomon [3] on Coxeter groups (finite groups of symmetries of R generated by reflections). A secondary purpose is to give more publicity to the fascinating special case of symmetric groups. This case was handled by Etienne whose recent paper [2] kindled my interest in this subject; I owe to Michelle Wachs my awareness of Solomon's earlier and more general theorem. The new proof is shorter and apparently more elementary than the proof of [3] (and the proof in the appendix to [3] due to Tits); nor does it reduce to Etienne's proof in the special case of the symmetric group. We begin by recalling some of the terminology of Coxeter groups. By definition they are finite groups of symmetries of n-dimensional real Euclidean space generated by those elements which are reflections in hyperplanes. Each reflecting hyperplane has two unit vectors r and — r orthogonal to it called roots and the set of roots has a reasonably canonical partition into positive roots and negative roots. The set of positive roots has a distinguished subset called the set of fundamental roots and the associated set of fundamental reflections (the Coxeter generators) also generates the group; with respect to this generating set the group has a very simple set of defining relations. Suppose that {T15 T2, ..., rm} is a set of Coxeter generators for a Coxeter group G associated with the set of fundamental roots {rx, r2,.. . , rm}. For any geG let X(g) denote the length of a minimal word in the generators which represents g. It is known [1, 6.1.1] that X(Zig) = X(g)± 1, with the positive sign if rtg is a positive root and the negative sign if rtg is a negative root. If S is any subset of {1, 2, ..., m) the signature class associated with S is the set
TL;DR: There is a one-to-one correspondence among switching classes of directed graphs, two-digraphs, and cyclic triple covers of the complete digraphs and the elements of the theory are these.
TL;DR: The soluble subgroups of maximal order of the symmetric, alternating, general and special linear groups are determined in this article, and they constitute just one conjugacy class, however, infinitely many exceptions.
Abstract: The soluble subgroups of maximal order of the symmetric, alternating, general and special linear groups are determined. Usually, they constitute just one conjugacy class. There are, however, infinitely many exceptions.
TL;DR: In this article, it was shown that the symmetric group of degree four is a solvable group, but it is not supersolvable, and it is shown that all the super-solvable groups are solvable.
Abstract: If G satisfies the permutizer condition, we call G a pc-group; clearly, all the super-solvable groups are pc-groups. It was shown in [1] that pc-groups of odd order were supersolvable. It is easy to verify that the symmetric group of degree four is a solvable pc-group, but it is not supersolvable. In general, solvable pc-groups
TL;DR: In this article, a generalization of the cycle notation for permutations is introduced for partial one-one transformations (charts) and notational representation theorems for charts that generalize those of permutations are given.
Abstract: A generalization of the cycle notation for permutations is introduced for partial one-one transformations (charts). Notational representation theorems for charts that generalize those of permutations are given. Notational multiplication of charts is developed and then applied to yield a transparent proof of Frobenius' result which bounds the idempotent in the cyclic subsemigroup. Lastly, the well known result that the structure of the cyclic subgroups of the finite symmetric groups is determined from combinations of disjoint cycles is generalized to the cyclic subsemigroups of the finite symmetric inverse semigroups.