We formulate quantum mechanics in the two-dimensional torus without using position operators. We define an algebra with only momentum operators and shift operators and construct irreducible representation of the algebra. We show that it realizes quantum mechanics of a charged particle in a uniform magnetic field. We prove that any irreducible representation of the algebra is unitary equivalent ...

For an embedding φ of the Cuntz algebra OM into ON , OM is identified with a subalgebra φ(OM) of ON . We construct irreducible representations of ON with continuous parameters by extending irreducible representations of OM . They are not unitarily equivalent to any generalized permutative representation, especially not to any permutative representation by Bratteli-Jorgensen and Davidson-Pitts. ...

The symplectic group branching algebra, B, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp2n−2(C) in each finite-dimensional irreducible representation of Sp2n(C). By describing on B an ASL structure, we construct an explicit standard monomial basis of B consisting of Sp2n−2(C) highest weight vectors. Moreover, B is known to carry a canonical ...

An action of an algebraic reductive groupG on an affine varietyM is calledmultiplicity-free if the multiplicity of any particular irreducible representation of G in the space C [M ] of regular functions on M is at most one: In [1], Kac provides a complete list of multiplicity-free actions for the case when G is a connected reductive algebraic group and M is a finite-dimensional vector space upo...

SupposeP is a parabolic subgroup ofGwith Levi factor M and σ = ⊗σv a cuspidal representation of M(A). Then Ind σ = ⊗vInd σv is a representation of G(A) which may not be irreducible, and may not even have a finite composition series. As usual an irreducible subquotient of this representation is said to be a constituent of it. For almost all v, Ind σv has exactly one constituent π◦ v containing t...

We show that the quantum Heisenberg group Hq(1) can be obtained by means of contraction from quantum SUq(2) group. Its dual Hopf algebra is the quantum Heisenberg algebra Uq(h(1)). We derive left and right regular representations for Uq(h(1)) as acting on its dual Hq(1). Imposing conditions on the right representation the left representation is reduced to an irreducible holomorphic representati...

Fix a prime p > 2. Let ρ : Gal(Q/Q) → GL2(I) be the Galois representation coming from a non-CM irreducible component I of Hida’s p-ordinary Hecke algebra. Assume the residual representation ρ̄ is absolutely irreducible. Under a minor technical condition we identify a subring I0 of I containing Zp[[T ]] such that the image of ρ is large with respect to I0. That is, Im ρ contains ker(SL2(I0) → SL2...

We have started this class by studying the representation theory of the symmetric group Sn over the complex numbers. We finish by giving a brief introduction to the representation theory of Sn over a field F of positive characteristic p. We will also establish a connection between the representations of ŝlp and those of FSn. This connection was one of motivations to consider Kac-Moody algebra a...

We classify, among the linear algebraic groups over a local field of characteristic zero, those that have the Haagerup property (also called a-(T)-menability). Our method relies essentially on a discussion on the existence of a subgroup isomorphic, up to a finite covering, to the semidirect product of SL2 by an irreducible representation, or a one-dimensional central extension of an even-dimens...

An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is (A) = 1 2 t(M) where M are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of the reducible fiel...