scholarly journals On the natural representation of the symmetric groups

1962 ◽  
Vol 5 (3) ◽  
pp. 121-136 ◽  
Author(s):  
H. K. Farahat
Keyword(s):  
Finite Number ◽  
Symmetric Group ◽  
Commutative Ring ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Unit Element ◽  
Natural Representation ◽  
Image Position ◽  
Obvious Manner

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,

scholarly journals Note on a paper of Tsuzuku

1964 ◽  
Vol 6 (4) ◽  
pp. 196-197
Author(s):  
H. K. Farahat
Keyword(s):  
Symmetric Group ◽  
Commutative Ring ◽  
Permutation Group ◽  
Matrix Representation ◽  
Invertible Element ◽  
Unit Element ◽  
Transitive Permutation Group ◽  
Natural Representation ◽  
Correct Proof ◽  
Permutation Matrices

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

scholarly journals The center of the wreath product of symmetric group algebras

2021 ◽  
Vol 31 (2) ◽  
pp. 302-322
Author(s):  
O. Tout ◽  
Keyword(s):  
Symmetric Group ◽  
Recent Result ◽  
Wreath Product ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Hyperoctahedral Group ◽  
Symmetric Group Algebras

We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.

Unitary Representations of Generalized Symmetric Groups

1969 ◽  
Vol 21 ◽  
pp. 28-38 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Unitary Representations ◽  
Complex Field ◽  
Permutation Matrices ◽  
Image Position

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

scholarly journals WHICH ALTERNATING AND SYMMETRIC GROUPS ARE UNIT GROUPS?

2013 ◽  
Vol 13 (03) ◽  
pp. 1350114 ◽  
Author(s):  
CHRISTOPHER DAVIS ◽  
TOMMY OCCHIPINTI
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Unit Group ◽  
Existence Results ◽  
Alternating And Symmetric Groups

We prove there is no ring with unit group isomorphic to Sn for n ≥ 5 and that there is no ring with unit group isomorphic to An for n ≥ 5, n ≠ 8. To prove the non-existence of such a ring, we prove the non-existence of a certain ideal in the group algebra 𝔽2[G], with G an alternating or symmetric group as above. We also give examples of rings with unit groups isomorphic to S1, S2, S3, S4, A1, A2, A3, A4, and A8. Most of our existence results are well-known, and we recall them only briefly; however, we expect the construction of a ring with unit group isomorphic to S4 to be new, and so we treat it in detail.

Vector Invariants of Symmetric Groups

1990 ◽  
Vol 33 (4) ◽  
pp. 391-397 ◽  
Author(s):  
H. E. A. Campbell ◽  
I. Hughes ◽  
R. D. Pollack
Keyword(s):  
Symmetric Group ◽  
Commutative Ring ◽  
Symmetric Groups ◽  
Free Module ◽  
Poincaré Series ◽  
Poincare Series ◽  
Direct Sum ◽  
Fixed Basis

AbstractLet M be a free module of rank n over a commutative ring R with unit and let Σn denote the symmetric group acting on a fixed basis of M in the usual way. Let Mm denote the direct sum of m copies of M and let S be the symmetric ring of Mm over R. Then each element of Σn acts diagonally on Mm and consequently on S; denote by Xn the subgroup of Gl(Mm) so defined. The ring of invariants, SΣn, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and SΣn have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.

scholarly journals Construction of Codes from Symmetric Groups

2019 ◽  
Vol 8 (4) ◽  
pp. 8658-8665
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Minimum Distance ◽  
Symmetric Groups ◽  
Semisimple Group ◽  
Group Codes ◽  
Complete Set

Let FSn be semisimple group algebra where Sn denotes the Symmetric group of degree n. We obtain the complete set of irreducible linear idempotents of the group algebra FSn. We also find the dimension and minimum distance of the group codes over the group S

On the transfer in the homology of symmetric groups

1978 ◽  
Vol 83 (1) ◽  
pp. 91-101 ◽  
Author(s):  
Daniel S. Kahn ◽  
Stewart B. Priddy
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Homotopy Theory ◽  
Symmetric Groups ◽  
Characterization Theorem ◽  
Group Cohomology ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Transfer Homomorphism ◽  
Image Position

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphismin group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

Proof that there are infinitely many modalities in Lewis's system S2

1940 ◽  
Vol 5 (3) ◽  
pp. 110-112 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Finite Number ◽  
Infinite Matrix ◽  
Image Position

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S2 is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S2 (which is the next stronger system than S2 discussed by Lewis).I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then

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