We study testing properties of functions on finite groups. First we consider functions of the form f : G → C, where G is a finite group. We show that conjugate invariance, homomorphism, and the property of being proportional to an irreducible character is testable with a constant number of queries to f , where a character is a crucial notion in representation theory. Our proof relies on represe...

Let F be a finite field or a local field of any characteristic. If A is a finite dimensional associative nilpotent algebra over F , the set 1 +A of all formal expressions of the form 1 +x, where x ∈ A, is a locally compact group with the topology induced by the standard one on F and the multiplication (1 + x) · (1 + y) = 1 + (x + y + xy). We prove a result conjectured by E. Gutkin in 1973: ever...

The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group Γ with an irreducible unitary representation ρ and satisfying u(γz) = ρ(γ)u(z). The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corre...

The Clebsch-Gordan formulas for d-function are studied for both nonunitary finite dimensional representation and unitary infinite dimensional representation of the SU (1, 1) group. It is shown that, in addition to the usual Clebsch-Gordan formulas for the above-mentioned two representations, there exists an additional type of Clebsch-Gordan fornmla connecting the nonunitary and unitary d-functi...

We explicitly construct an irreducible four-dimensional unitary representation of the Lie group SU (2) and use it for the construction of high-rate unitary space-time codes for four transmit antennas. The tools we use are invariant integration on the group to unitarize the representation, and dense spherical codes on the three-dimensional sphere. Our construction calls for the design of such sp...

If the locally compact abelian group G has a finitedimensional unitary irreducible projective representation with factor system co (i.e. G has an a>-rep), then a subgroup G(co) is defined which fulfils three roles. First, the square-root of the index of G(u)) in G is the dimension of every co-rep. Secondly, the co-reps of G can be labelled by the dual group of G(co), up to unitary equivalence. ...

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous spac...

If G is a Lie group, H ⊂ G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ ∈ ig∗ to be in the wave front set of IndH τ . In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a reductive Lie group of Harish-Chandra class and π is a unitary representation of G that is weakly containe...