The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F , for which A is an irreducible component of V(F ), a generic linear space L of complementary dimension to A, and a numerical approximation to the set of witness points, L ∩A. Given F , methods exist for computing a numeri...

Highest weight representations of Uq(su(1, 1)) with q = expπi/N are investigated. The structures of the irreducible hieghesat weight modules are discussed in detail. The Clebsch-Gordan decomposition for the tensor product of two irreducible representations is discussed. By using the results, a representation of SL(2,R) ⊗ Uq(su(2)) is also presented in terms of holomorphic sections which also ha...

With each resonance of the Laplacian acting on compactly supported sections a homogeneous vector bundle over Riemannian symmetric space non-compact type, One can associate residue representation. The purpose this paper is to study them. assumed have rank-one but irreducible representation {\tau} K defining arbitrary. We give an algorithm that aims at determining if these representations are irr...

Frobenius began the study of representation theory and character theory of the symmetric groups Sn at the turn of the century [4]. In this study, He showed that there is one irreducible representation πλ of Sn corresponding to each partition λ of n and this correspondence gives all of irreducible representations of Sn up to equivalency. After this work, he also began the study of representation...

Symplectic branching involves multiplicities greater than one. Let L be the n-fold product of SL(2, C). We prove that each multiplicity space that arises in the restriction of an irreducible representation of Sp(n, C) to Sp(n − 1, C) is naturally an irreducible L-module. As an application we obtain a Gelfand-Zeitlin type basis for all irreducible finite dimensional representations of Sp(n, C).

When a three-dimensional object moves relative to an observer, a change occurs on the observer’s image plane and in the visual representation computed by a learned model. Starting with the idea that a good visual representation is one that transforms linearly under scene motions, we show, using the theory of group representations, that any such representation is equivalent to a combination of t...

We study the regular representation ρζ of the single-fermion algebra Aζ , i.e., c2 = c+2 = 0, cc+ + c+c = ζ 1, for ζ ∈ [0, 1]. We show that ρ0 is a four-dimensional nonunitary representation of A0 which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ρ0 is the minimal faithfully irreducible representation of A0 in the sense that every faithful repres...

Tensor operators in graded representations of Z2−graded Hopf algebras are defined and their elementary properties are derived. WignerEckart theorem for irreducible tensor operators for Uq[osp(1 | 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra Uq[osp(1 | 2)] are considered. The reduced matrix elements for the irreducible tensor operators are c...

The irreducible bases in the icosahedral group space are calculated explicitly by reducing the regular representation. The symmetry adapted bases of the system with I or Ih symmetry can be calculated easily and generally by applying those irreducible bases to wavefunctions of the system, if they are not vanishing. As examples, the submatrices of the Hückel Hamiltonians for Carbon-60 and Carbon-...

We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced...