scholarly journals The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

2020 ◽  
pp. 2150209
Author(s):  
Nanying Yang ◽  
Alexey M. Staroletov
Keyword(s):  
Symmetric Groups ◽  
Irreducible Representations ◽  
Alternating Groups ◽  
Minimal Polynomials ◽  
Alternating And Symmetric Groups

Denote the alternating and symmetric groups of degree [Formula: see text] by [Formula: see text] and [Formula: see text], respectively. Consider a permutation [Formula: see text], all of whose nontrivial cycles are of the same length. We find the minimal polynomials of [Formula: see text] in the ordinary irreducible representations of [Formula: see text] and [Formula: see text].

scholarly journals Generators for Alternating and Symmetric Groups

1971 ◽  
Vol 12 (1) ◽  
pp. 63-68 ◽  
Author(s):  
I. M. S. Dey ◽  
James Wiegold
Keyword(s):  
Free Product ◽  
Modular Group ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Generating Set ◽  
Alternating And Symmetric Groups

Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newman's interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF(2,p) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

On the generating graphs of symmetric groups

2018 ◽  
Vol 21 (4) ◽  
pp. 629-649 ◽  
Author(s):  
Fuat Erdem
Keyword(s):  
Symmetric Groups ◽  
Alternating Groups

AbstractLet {S_{n}} and {A_{n}} be the symmetric and alternating groups of degree n, respectively. Breuer, Guralnick, Lucchini, Maróti and Nagy proved that the generating graphs {\Gamma(S_{n})} and {\Gamma(A_{n})} are Hamiltonian for sufficiently large n. However, their proof provided no information as to how large n needs to be. We prove that the graphs {\Gamma(S_{n})} and {\Gamma(A_{n})} are Hamiltonian provided that {n\geq 107}.

scholarly journals Specht modules for finite reflection groups

1995 ◽  
Vol 37 (3) ◽  
pp. 279-287 ◽  
Author(s):  
S. HalicioǦlu
Keyword(s):  
Present Author ◽  
Symmetric Groups ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Young Tableaux ◽  
Reflection Groups ◽  
Specht Modules ◽  
Irreducible Modules ◽  
Original Approach

Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A. Young, and of irreducible modules, called Specht modules, due to W. Specht, for the symmetric groups Sn which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [13]). James [12] extended these ideas to construct irreducible representations and modules over an arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to cover the Weyl groups of type Bn. In [14] Morris described a possible extension of James' work for Weyl groups in general. Later, the present author and Morris [8] gave an alternative generalisation of James' work which is an extended improvement and extension of the original approach suggested by Morris. We now give a possible extension of James' work for finite reflection groups in general.

The factorisation of the alternating and symmetric groups

1980 ◽  
Vol 175 (2) ◽  
pp. 171-179 ◽  
Author(s):  
James Wiegold ◽  
Alan G. Williamson
Keyword(s):  
Symmetric Groups ◽  
Alternating And Symmetric Groups

scholarly journals On alternating and symmetric groups which are quasi OD-characterizable

2017 ◽  
Vol 16 (04) ◽  
pp. 1750065 ◽  
Author(s):  
Ali Reza Moghaddamfar
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Symmetric Groups ◽  
Degree Pattern ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

REPRESENTATIONS OF THE SYMPLECTIC ROOK MONOID

2008 ◽  
Vol 18 (05) ◽  
pp. 837-852 ◽  
Author(s):  
ZHENHENG LI ◽  
ZHUO LI ◽  
YOU'AN CAO
Keyword(s):  
Weyl Group ◽  
Symmetric Groups ◽  
Triangular Matrix ◽  
Character Table ◽  
Irreducible Representations ◽  
Rook Monoid ◽  
Algebraic Description ◽  
Practical Algorithm ◽  
Upper Triangular Matrix ◽  
User Friendly

In this paper, we concern representations of symplectic rook monoids R. First, an algebraic description of R as a submonoid of a rook monoid is obtained. Second, we determine irreducible representations of R in terms of the irreducible representations of certain symmetric groups and those of the symplectic Weyl group W. We then give the character formula of R using the character of W and that of the symmetric groups. A practical algorithm is provided to make the formula user-friendly. At last we show that the Munn character table of R is a block upper triangular matrix.

Quasi-permutation representations of alternating and symmetric groups*

2007 ◽  
Vol 27 (2) ◽  
pp. 297-300
Author(s):  
Behravesh Houshang ◽  
Hossein Jafari Mohammad
Keyword(s):  
Symmetric Groups ◽  
Alternating And Symmetric Groups
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