Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

Equality of Decomposable Symmetrized Tensors

1975 ◽  
Vol 27 (5) ◽  
pp. 1022-1024 ◽  
Author(s):  
Russell Merris
Keyword(s):  
Linear Operator ◽  
Tensor Product ◽  
Vector Space ◽  
Symmetric Group ◽  
Dimensional Vector ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Image Position

Let V be an n-dimensional vector space over the field F. Let ꕕm V be the rath tensor power of V. If ᓂ ∈ Sm, the symmetric group, there exists a linear operator P (ᓂ1) on ꕕm V such thatfor all x1, … , xm ∈ V. (Here, x1 ꕕ … ꕕ xm denotes the decomposable tensor product of the indicated vectors.) If c is any function of Sm taking its values in F, we define

scholarly journals Real Subspaces of a Quaternion Vector Space

1978 ◽  
Vol 30 (6) ◽  
pp. 1228-1242 ◽  
Author(s):  
Vlastimil Dlab ◽  
Claus Michael Ringel
Keyword(s):  
Vector Space ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Image Position ◽  
Real Subspace

If UR is a real subspace of a finite dimensional vector space VC over the field C of complex numbers, then there exists a basis ﹛e1, … , en﹜ of VG such that

Rational Tensor Representations of Hom(V, V) and an Extension of an Inequality of I. Schur.

1972 ◽  
Vol 24 (4) ◽  
pp. 686-695 ◽  
Author(s):  
Marvin Marcus ◽  
William Robert Gordon
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Linear Operators ◽  
Inner Product ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Semi Group ◽  
Rational Representation ◽  
Dimensional Vector Space ◽  
Tensor Representations

Let V be an n-dimensional vector space over the complex numbers equipped with an inner product (x, y), and let (P, μ) be a symmetry class in the mth tensor product of V associated with a permutation group G and a character χ (see below). Then for each T ∊ Hom (V, V) the function φ which sends each m-tuple (v1, … , vm) of elements of V to the tensor μ(TV1, … , Tvm) is symmetric with respect to G and x, and so there is a unique linear map K(T) from P to P such that φ = K(T)μ.It is easily checked that K: Hom(V, V) → Hom(P, P) is a rational representation of the multiplicative semi-group in Hom(V, V): for any two linear operators S and T on VK(ST) = K(S)K(T).Moreover, if T is normal then, with respect to the inner product induced on P by the inner product on V (see below), K(T) is normal.

scholarly journals A Norm Inequality for Linear Transformations

1961 ◽  
Vol 4 (3) ◽  
pp. 239-242
Author(s):  
B.N. Moyls ◽  
N.A. Khan
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Hermitian Operator ◽  
Norm Inequality ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Image Position ◽  
Ky Fan

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then1

scholarly journals Rank k Vectors in Symmetry Classes of Tensors

1976 ◽  
Vol 19 (1) ◽  
pp. 67-76 ◽  
Author(s):  
Ming-Huat Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Multiplicative Group ◽  
Vector Spaces ◽  
Linear Character ◽  
Symmetry Classes ◽  
Multilinear Function ◽  
Image Position

Let F be a field, G a subgroup of Sm, the symmetric group of degree m, and χ a linear character on G, i.e., a homomorphism of G into the multiplicative group of F. Let V1,...,Vm be vector spaces over F such that Vi = Vσ(i) for i=1,…,m and for all σ∈G. If W is a vector space over F, then a m-multilinear function is said to be symmetric with respect to G and χ iffor any σ ∊ G and for arbitrary xi ∊ Vi.

Anticommuting Linear Transformations

1961 ◽  
Vol 13 ◽  
pp. 614-624 ◽  
Author(s):  
H. Kestelman
Keyword(s):  
Quantum Mechanics ◽  
Vector Space ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Pauli Matrices ◽  
Image Position

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices(1)where the B1,r are the Pauli matrices(2)(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), isTheorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfyingthen there is a matrix T such thatand T is unique apart from an arbitrary numerical multiplier.

scholarly journals Existence of continuous right inverses to linear mappings in finite-dimensional geometry

2020 ◽  
Author(s):  
Christer Oscar Kiselman ◽  
Erik Melin
Keyword(s):  
Vector Space ◽  
Convex Subset ◽  
Compact Convex ◽  
Linear Mapping ◽  
Compact Convex Subset ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Abstract A linear mapping of a compact convex subset of a finite-dimensional vector space always possesses a right inverse, but may lack a continuous right inverse, even if the set is smoothly bounded. Examples showing this are given, as well as conditions guaranteeing the existence of a continuous right inverse.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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