Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

Extendible elements of the alternating groups

An element?σ?of A_n , the Alternating group of degree n, is extendible in S_n , the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Direct products of automorphism groups of graphs

In this article we study the product action of the direct product of automorphism groups of graphs. We generalize the results of Watkins [J. Combin Theory 11 (1971), 95–104], Nowitz and Watkins [Monatsh. Math. 76 (1972), 168–171] and W. Imrich [Israel J. Math. 11 (1972), 258–264], and we show that except for an infinite family of groups S_n × S_n, n≥2 and three other groups D₄ × S₂, D₄ × D₄ and S₄ × S₂ × S₂, the direct product of automorphism groups of two graphs is itself the automorphism group of a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 26–36, 2009     

The young diagrams of a pair of irreducible characters of S n with the same zero set on S n ?

In studying the pairs of irreducible characters of the symmetric group S _n with the same zero set on A _n or S _n ∖ A _n (as well as the pairs of irreducible characters with the same zero set on the alternating group A _n ), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem that considerably generalizes two previous results of frequent use in this direction. Original Russian Text Copyright ? 2008 Belonogov V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00148) and the RFBR-NSFC (Grant 05-01-39000). __________ Ekaterinburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 5, pp. 992–1006, September–October, 2008.     

The Minimal Number of Cyclic Generators of the Symmetric and Alternating Groups

The aim of this note is to find the minimal number of generators of the symmetric group S _n and alternating group A _n , when the generators are cycles of length at most k. The approach is constructive.     

A representation on the labeled rooted forests

We consider the conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. By interpreting these objects in terms of labeled rooted forests, we give a characterization of the labeled rooted trees whose S_n orbit afford the sign representation. Applications to rook theory are offered.     

Linear Representations of the Braid Groups of Some Manifolds

In this paper we prove that the braid group B_n(S²) of 2-sphere, mapping class group M(0,n) of the n-punctured 2-sphere and the braid group B₃(P²) of the projective plane are linear. Partially supported by the Russian Foundation for Basic Research (grant number 02-01-01118).Mathematics Subject Classifications (2000) 20F28, 20F36, 20G35.     

Involutive automorphism of symmetric groups

Let (𝔖_n,S) be a Coxeter system of the symmetric group, we show that the set of automorphisms of 𝔖_n which are involutions and leave S stable is a finite group of order less than 3.     

The distinguishing number of the direct product and wreath product action

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted D _G(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ≀_Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S _m × S _n on [m] × [n].     

Finite Groups with S-permutable n-maximal Subgroups

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n?1(G) of order 4 are S-permutable in G.     

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

Recognition of some symmetric groups by the set of the order of their elements

For a finite group G, let π_e(G) be the set of order of elements in G and denote S _n the symmetric group on n letters. We will show that if π_e(G ) = π_e(H), where H is S _p or S _p+1 and p is a prime with 50 < p < 100, then G ≅ H. This revised version was published online in August 2006 with corrections to the Cover Date.     

Polynomial Invariants of Certain Pseudo-Symplectic Groups Over Finite Fields of Characteristic Two

Let F _q be a finite field of characteristic two, S be a nonsingular non-alternate symmetric matrix over F _q and Ps _n (F _q , S) be the associated pseudo-symplectic group. Let Ps _n (F _q , S) act linearly on the polynomial ring F _q [x ₁,…, x _n ]. In this note, we find an explicit set of generators of the ring of invariants of Ps _n (F _q , S) for n = 2, 4 and 2ν +1. In particular, the results assert that the ring of invariants of Ps ₄(F _q , S) is not a polynomial algebra but is an example of hypersurface and the ring of invariants of Ps _2ν+1(F _q , S) is a complete intersection.     

EXCEPTIONAL SEQUENCES FOR QUIVERS OF DYNKIN TYPE

Let kQ be the path algebra of a quiver Q without oriented cycles with n vertices. An indecomposable kQ-module without self-extensions is called exceptional. The braid group B _n with n ? 1 generators acts naturally on the set of complete exceptional sequences. Crawley-Boevey (Proceedings of ICRA VI, Carleton-Ottawa, 1992) and Ringel (Contemp. Math. 1994, 171, 339–352) have pointed out that this action is transitive. The number of complete exceptional sequences for kQ representation finite will be computed here and it is shown to be independent of the orientation of the arrows of the quiver Q. The factor group of the braid group which acts freely on the set of complete exceptional sequences can be regarded as a subgroup of the symmetric group S _{? n} , where ? _n is the number of complete exceptional sequences of the algebra kQ. This group is known for certain special types of quivers. Some other interesting relations of the acting group will be given.     

Equivariant Reduction to Torus of a Principal Bundle

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k. Let E _G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E _G admits a reduction of structure group to a maximal torus TG. We give a necessary and sufficient condition for the existence of a T-reduction of E _G which is left invariant by the action of S on E _G .     

Rapidly Mixing Random Walks and Bounds on Characters of the Symmetric Group

We investigate mixing of random walks on S _n and A _n generated by permutations of a given cycle structure. The approach follows methods developed by Diaconis, which requires certain estimates on characters of the symmetric group and uses combinatorics of Young tableaux. We conclude with conjectures and open problems.     

A stability result for balanced dictatorships in Sn

We prove that a balanced Boolean function on S_n whose Fourier transform is highly concentrated on the first two irreducible representations of S_n, is close in structure to a dictatorship, a function which is determined by the image or pre‐image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on S_n generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [6] deals with Boolean functions of expectation O(1/ n) whose Fourier transform is highly concentrated on the first two irreducible representations of S_n. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point‐stabilizers. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 494–530, 2015     

Commutative Schur Rings of Maximal Dimension

A commutative Schur ring over a finite group G has dimension at most s _G  = d ₁ + … +d _r , where the d _i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 ⁿ ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.     

Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group