QUASI-PERMUTATION REPRESENTATIONS OF ALTERNATING AND SYMMETRIC GROUPS

The quantities c（G）, q（G） and p（G） for finite groups were defined by H. Behravesh. In this article, these quantities for the alternating group An and the symmetric group Sn are calculated. It is shown that c（G） = q（G） = p（G） = n,when G = An or Sn.

A Note on the Signature Representations of the Symmetric Groups

For a partition λ and a prime p,we prove a necessary and sufficient condition for there to exist a composition δ such that δ can be obtained from λ after rearrangement and no partial sums of δ are divisible by p.To demonstrate why we are interested in the question,we compute some signed p-Kostka numbers.

Double Derangement Permutations

Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .

Integral Cayley Graphs over Finite Groups

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group(s)is integral.In particular,a Cayley graph of a 2-group generated by a normal set of involutions is integral.We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral.We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form(k i j)with fixed k,a s{-n+1,1-n+1,2^2-n+1,...,(n-1)2-n+1}.

Pfaffians and Representations of the Symmetric Group

Pfaffians of matrices with entries z[i, j]/（xi ＋ xj）, or determinants of matrices with entries z[i, j]/（xi - xj）, where the antisymmetrical indeterminates z[i, j] satisfy the Pliicker relations, can be identified with a trace in an irreducible representation of a product of two symmetric groups. Using Young＇s orthogonal bases, one can write explicit expressions of such Pfaffians and determinants, and recover in particular the evaluation of Pfaffians which appeared in the recent literature.

Explicit Representations for Local Lagrangian Numerical Differentiation

It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.

Some Remarks on the Symmetry of Self-Conjugate Minimal Surfaces in R^3

Symmetry properties of self-conjugate minimal surfaces,i.e.minimal surfaces which are congruent with their conjugate ones in R^3 are studied.

A New Class for Large Sets of Almost Hamilton Cycle Decompositions

A k-cycle system of order v with index A, denoted by CS（v, k, λ）, is a collection A of k-cycles （blocks） of Kv such that each edge in Kv appears in exactly λ blocks of A. A large set of CS（v, k, λ）s is a partition of the set of all k-cycles of Kv into CS（v, k, λ）s, and is denoted by LCS（v, k, λ）. A （v - 1）-cycle in K, is called almost Hamilton. The completion of the existence problem for LCS（v, v- 1,λ） depends only on one case： all v ≥ 4 for λ=2. In this paper, it is shown that there exists an LCS（v,v - 1,2） for all v ≡ 2 （mod 4）, v ≥ 6.

A Virtual Character and Nonzero Kronecker Coefficients for Self-conjugate Partitions

We generalize Regev's results on a virtual character of the symmetric group S_(n),that is,an integer-combination of some irreducible characters.Suppose that λ and μ are integer partitions of n.For the associated irreducible character χ^(λ) of S_(n),when χ^(λ)(μ)≠0,we find another partition τ related to λ such that χ^(τ)(μ)≠0 by the virtual character.Applying this result,we obtain a class of nonzero Kronecker coefficients by Pak et al.'s character criterion.Moreover,we discuss the effectiveness of Pak et al.'s character criterion bya concreteexample.

On Tensor Spaces for Rook Monoid Algebras

Let m, n ∈ N, and V be an m-dimensional vector space over a field F of characteristic 0. Let U = F ＋ V and Rn be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U^×n in FRn, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of U^×n in FR^n.