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Analysis on Locally Compact Groups
Published in Hugo D. Junghenn, Principles of Analysis, 2018
The direct product of groups G and H is the group G×H $ G\times H $ with multiplication (a,b)(x,y)=(ax,by),a,x∈G,b,y∈H. $$ (a,b)(x,y) = (ax,by), \ \ a, x \in G, \ \ b, y \in H. $$
Fast Fourier Transforms for Motion Groups
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Given the pairs g = (a, A) and h = (r, R) where a, r ∈ ℝD and A, R ∈ SO(D), if no group law is specified, there is no natural definition of the Fourier transform. That is, g and h could be elements of the semi-direct product group SE(D) or the direct product group ℝD × SO(D). And while the natural choice for the definition of a group Fourier transform is based on the IURs of that group, the similarities between SE(D) and ℝD × SO(D) are worth exploiting because the IURs of the former are infinite-dimensional, while those of the latter are finite dimensional.
The Weak Interaction in the Framework of Grand Unification Theories
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
That the GWS theory still contains two coupling constants comes about because the theory is based on two independent types of gauge transformations, the SU(2)L transformations and the U(1) transformations. The full gauge group GGWS is the direct product of two subgroups: () GGWS=SU(2)L⊗U(1)
On extending and optimising the direct product decomposition
Published in Molecular Physics, 2019
The point group of a molecule is composed of the set of symmetry operations such as reflections, rotations, inversion, etc. that leave the molecular framework unchanged. These operations form a group in the mathematical sense, with multiplication defined as the composition of symmetry operations. When an object (such as a molecular orbital) is subjected to one of these operations, its behaviour can be described by its character, that is the overlap of the object before and after the operation. The collection of the characters for each operation gives the representation. For combinations of objects, such as a charge distribution defined by a pair of orbitals, the individual characters of the overall object are the product of the individual characters. Taking the representations as a whole, this element-wise multiplication to give a new combined representation is called the direct product (⊗), not to be confused with the outer or direct product used in other areas of mathematics. In order to describe and classify arbitrary representations, the irreducible representations (irreps) of the group form a unique basis: the way in which any particular object transforms (its representation) can be uniquely decomposed into a linear combination of the irreps. Each group has an irrep called the totally-symmetric irrep which has characters of unity. In order for an integral to be non-zero, the combined representation of the integrands must include (have non-zero projection onto) this totally-symmetric irrep. Individual components of the integrand (integrals, amplitudes, etc.) are generally chosen to be symmetry-adapted to a particular irrep, meaning that e.g. the direct product of the orbital representations for a given integral or amplitude tensor element must include the chosen overall irrep in order for that element to be non-zero. This sparsity is the origin of reduced computational cost when using point group symmetry.