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Functional Equations on Affine Groups
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
All double cosets with respect to H form a partition on G, and the factor set with respect to the corresponding equivalence relation is the double coset space G//H. Accordingly, the canonical mapping Φ:G→G//H is defined by Φ(g)=HgH
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Membership in a double coset is an equivalence relation on G. That is, G is partitioned into disjoint double cosets, and for H < G and K < G either Hg1K ∩ Hg2K = ø or Hg1K = Hg2K.
On extending and optimising the direct product decomposition
Published in Molecular Physics, 2019
However, in order to facilitate the identification of components leading to either symmetric or non-symmetric integrands, and also in order to produce a form leading to the smallest number of non-zero contributions, the basic building blocks of the calculation, most importantly the atomic and molecular orbitals, may be chosen to transform as irreducible representations (these being a special and useful class of representations) of the point group, a process called symmetry-adaptation. Transformation of integrals from the atomic orbital (AO) basis into the symmetry orbital (SO) basis using explicit transformation matrices has also been considered for some time. However, this approach requires computation of the entire AO integral set which may be highly redundant. Pitzer may have been the first to suggest an approach to generate SO integrals directly from the non-redundant set of atomic integrals [7], saving both time and memory space. Later approaches based on symmetry group generators [8] or the double coset decomposition [9,10] improved and formalised the generation of SO integrals, in particular for non-degenerate point groups (D2h and subgroups), although generation of SO integrals for degenerate groups (non-Abelian groups as well as separably-degenerate [11,12] groups such as Cn) is possible using the latter approach. Of course, even without the explicit construction of symmetry orbitals, the molecular orbitals (MOs) transform as irreducible representations of the molecular point group, leading the symmetry blocking of the MO coefficients and MO operator matrices. For correlated (post-SCF) calculations, this means that the source of the symmetry-adapted MO basis is not terribly important, and a symmetry-adapted basis may generally be taken for granted.