Showing papers on "Symmetric group published in 1992"

Random walk in a Weyl chamber

Abstract: The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge x 1 ≥x 2 ≥...≥x n (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps

The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group

Abstract: A combinatorial bijection is given between pairs of permutations in S n the product of which is a given n -cycle and two-coloured plane edge-rooted trees on n edges, when the numbers of cycles in the disjoint cycle representations of the permutations sum to n + 1 Thus the corresponding connection coefficient for the symmetric group is determined by enumerating these trees with respect to appropriate characteristics This is extended to the case of m -tuples of permutations in S n the product of which is a given n -cycle, in which the combinatorial objects replacing trees are cacti of m -gons

Analysis of Top To Random Shuffles

Abstract: A deck of n cards is shuffled by repeatedly taking off the top m cards and inserting them in random positions. We give a closed form expression for the distribution after any number of steps. This is used to give the asymptotics of the approach to stationarity: for m fixed and n large, it takes shuffles to get close to random. The formulae lead to new subalgebras in the group algebra of the symmetric group.

Quotients of Coxeter Complexes and P-Partitions

Abstract: Let $W$ be a finite reflection group acting on R$\sp n$. As $W$ preserves the unit sphere S$\sp{n-1}$, for any subgroup $G\ \subseteq\ \ W$, there is a quotient S$\sp{n-1} /G$ of this sphere under the action of $G$. We study the combinatorial (and topological) structure of these quotients as certain kinds of cell complexes (balanced simplicial posets). In particular, we give sufficient conditions on $G$ for the quotient to be Cohen-Macaulay or Gorenstein over a field $k$, and a simple characterization of those $G$ for which the quotient is a pseudomanifold, and when it is orientable as a pseudomanifold. We then look at quotients for particular classes of subgroups $G$, namely reflection subgroups, alternating subgroups of reflection subgroups, and their diagonal embeddings in the product groups $W\sp r$. For these groups, we show that the quotient is always partitionable, that in some cases it is shellable, and when shellable it is either a sphere or a disk. For all of these groups, the partitioning yields combinatorial interpretations for certain non-negative integers $\beta\sb J$ associated to the quotient known as the type-selected Mobius invariants. Applications to calculating invariant polynomials of permutation groups and their Hilbert series (as developed by Garsia and Stanton (GS)) are discussed. Our methods require an extension of some of the theory of P-partitions, and multi-partite P-partitions from the symmetric group $S\sb n$ to other finite reflection groups. In particular, for the hyperoctahedral group $B\sb n$, we work out analogues to almost all of the standard $P$-partition results. This yields hyperoctahedral analogues for the connection between posets and distributive lattices. These methods also suggest a new approach and generalization to the Neggers-Stanley conjecture. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690).

Coset Enumeration in Groups and Constructions of Symmetric Designs

Abstract: Publisher Summary This chapter discusses coset enumeration in groups and constructions of symmetric designs. It presents a group in terms of generators and relations so that each point or block stabilizer, can be expressed with the same generators. The Coxeter–Todd coset enumeration method with respect to the subgroup gives the number of cosets and also gives the permutation representation of group with respect to the (right) cosets of the subgroup. The corresponding programs have been made by Hrabe De Angelis for each stabilizer subgroup. To construct the design means only to put together all these permutation representations according to the orbit structure matrix. In JANKO–TRAN a symmetric design was constructed whose full automorphism group is discussed.

Algebraic properties of cryptosystem PGM

Abstract: In the late 1970s Magliveras invented a private-key cryptographic system calledPermutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, calledlogarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ|G| for a given finite permutation groupG. In this paper we prove several algebraic properties of PGM. We show that the set of PGM transformations ℐG is not closed under functional composition and hence not a group. This set is 2-transitive on ℤ|G| if the underlying groupG is not hamiltonian and not abelian. Moreover, if the order ofG is not a power of 2, then the set of transformations contains an odd permutation. An important consequence of these results is that the group generated by the set of transformations is nearly always the symmetric group ℒ|G|. Thus, allowing multiple encryption, any permutation of the message space is attainable. This property is one of the strongest security conditions that can be offered by a private-key encryption system.

The one-round functions of the DES generate the alternating group

Abstract: In each of the 16 DES rounds we have a permutation of 64-bitblocks. According to the corresponding key-block there are 248 possible permutations per round. In this paper we will prove that these permutations generate the alternating group. The main parts of the paper are the proof that the generated group is 3-transitive, and the application of a result from p. J. Cameron based on the classification of finite simple groups. A corollary concerning n-round functions generalizes the result.

Representations and traces of the Hecke algebras Hn(q) of type An−1

Abstract: The notion of the connectivity class of minimal words in the algebra Hn(q) is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented. Guided by these results, the connection between the Ocneanu trace on Hn(q) and Schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of Hn(q). They appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions. Their evaluation involves the use of the Littlewood–Richardson rule, which is proved to apply to Hn(q) just as it does to Sn. Both representation matrices and characters are tabulated.

A generalization of hadamard's inequality

Abstract: If A = (ai,j ) is an n × n complex matrix then h(A) denotes the product of the diagonal entries of A, and if λ is a partition of n then [λ](A) is defined by where Sn denotes the symmetric group of degree n and {λ} is the ordinary irreducible character of Sn corresponding to λ. Let deg λ denote the degree of {λ}. In this paper we investigate which partitions λ of n satisfy the inequality for all n × n Hermitian positive semi-definite matrices A. When λ = (1 n ) the inequality was proved by J. Hadamard. In this paper we show, amongst other things, that the inequality holds for: .

Symmetry classes of tensors associated with certain groups

Abstract: We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with certain subgroups of the full symmetric group The dimensions of these symmetry classes of tensors are also given

The framed braid group and 3-manifolds

Abstract: . The framed braid group on n strands is defined to be a semidirectproduct of the braid group B„ and Z" . Framed braids represent 3-manifoldsin a manner analogous to the representation of links by braids. Consider twoframed braids equivalent if they represent homeomorphic 3-manifolds. Themain result of this paper is a Markov type theorem giving moves that generatethis equivalence relation. In this paper the group of framed braids $n is introduced. This group issimilar to the braid group and is an initial attempt to understand 3-manifoldsin a manner analogous to the braid approach to links in the 3-sphere. The maintheorem describes the equivalence relations on \J^LX 5« that yields the set of 3-manifolds. Framed braids. Let Bn denote the braid group with generators ox, oj, ... , 1; (2) OiOi+xOi = Oi+xOiOi+x. The geometric braid ai is shown in Figure 1.Let X„ denote the symmetric group acting on {1, 2, ... , zz} . Let n : B„ -+Z„ be the quotient map sending a, to the transposition (j, z'+l). The kernel of?i is the pure braid group denoted by P„ . Bn acts on {1, 2, ... , zz} throughn, i.e., aii) = zt((r)(z) for a £ Bn. This paper follows the convention that thesymmetric group acts from the right so that (ot)(z) = t(<7(z')) for a, t £ B„ .Definition. The framed braid group #„ is the group generated by ax, a2, ... ,<7„_i , tx, I2, ... , tn with the relations (1), (2) and additional relations

The Poset of Conjugacy Classes and Decomposition of Products in the Symmetric Group

Abstract: The action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

On Schur's Q -functions and the Primitive Idempotents of a Commutative Hecke Algebra

Abstract: Let Bn denote the centralizer of a fixed-point free involution in the symmetric group S2n. Each of the four one-dimensional representations of Bn induces a multiplicity-free representation of S2n, and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of Sn.

Casimir operators, duality, and the point groups

Abstract: In atomic or nuclear shell theory the quadratic Casimir operators can be used to evaluate the expectation values of two‐body interactions among equivalent members of a shell. Racah has given closed expressions for these Casimir operators for all the groups in the Racah chain. In this paper an alternate derivation of these closed expressions based on duality with the symmetric group is given. The derivation allows extension of these expressions to situations in which equivalent spins are in an environment, such as a point group, for which the two‐body interactions separate into distinct sets. Possible application to NMR systems with spin ≥1/2 are discussed.

Bases and decomposition numbers of finite groups

Abstract: G. The minimal cardinality of a basis of G is denoted by r(G). A family of finite groups ,3 is well-based if there exists a constant c such that r(G) < c [GI 1/z for each G ~ .3. The problem of estimating r(G) for cyclic groups was first proposed by I. Schur and various bounds were obtained by Rohrbach [7], Moser [5], St6hr [9], Klotz [3] and others. Bases for arbitrary groups were dealt by Rohrbach [8] and lately by Bertram and Herzog [1] and Nathanson [6]. In [8] Rohrbach showed that the class of abelian groups with a bounded number of generators is well-based. He also mentioned that the class of solvable groups which possess a series of a bounded length with cyclic factors is well- based. In [1] Bertram and Herzog showed that the families of the nilpotent groups, as well as the families of the alternating and symmetric groups, are well-based. In [6] Nathanson showed that r(G) < 2(IGI loglG[) 1/2 + 2 for every finite group G of order n. In this paper we prove that the family of all finite groups is well-based, with r (G) < ~IGI

A complete classification of symmetric (31, 10, 3) designs

Abstract: In recent years several authors have determined all symmetric (31, 10, 3) designs with a nontrivial automorphism. Here we describe an algorithm for the generation by computer of all symmetric (31, 10, 3) designs and find that there are precisely 151 such nonisomorphic designs. Of these, 107 have a trivial automorphism group.

On the diameter of random Cayley graphs of the symmetric group

Abstract: Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

Hook formula and related identities

Abstract: A new proof of the hook formula for the dimension of representations of the symmetric group is given with the help of identities which are of independent interest. A probabilistic interpretation of the proof and new formulas relating the parameters of the Young diagrams are given.

Structure forest and composition factors for small base groups in nearly linear time

Abstract: A base of a permutation group G is a subset B of the permutation domain such that only the identity of G fixes B pointwise. The permutation representations of important classes of groups, including all finite simple groups other than the alternating groups, admit O(log n) size bases, where n is the size of the permutation domain. Groups with very small bases dominate the work on permutation groups within computational group theory.We use the “soft” version of the big-O notation introduced by [BLSI]: for two functions f(n), g(n), we write f(n)=O˜(g(n)) if for some constants c, C > 0, we have f(n) ≤ Cg(n) logcn.We address the problems of finding structure trees and composition factors for permutation groups with small (O˜(1) size) bases. For general permutation groups, a method of Atkinson will find a structure tree in O(n2) time. We give an O˜(n) algorithm for the small base case. The composition factor problem was first shown to have a polynomial time solution by Luks [Lu], and recently Babai, Luks, Seress [BLS2] gave an O˜(n3) algorithm. The [BLS2] algorithm takes T(n3) time even in the small base case. We overcome several quadratic and cubic bottlenecks in the [BLS2] algorithm to give an O˜(n) Monte Carlo algorithm for the small base case. In addition, we show that the center of a small base group can be found in time O˜(n).