On the realization of the Gelfand character of a finite group as a twisted trace

We show that the Gelfand character χ G \chi_{G} of a finite group 𝐺 (i.e. the sum of all irreducible complex characters of 𝐺) may be realized as a “twisted trace” g ↦ Tr ⁡ ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ⁢ ( G ) L^{2}(G) , where ( L 2 ⁢ ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ⁢ ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ⁢ ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ⁡ ( L ) \operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that ( L 2 ⁢ ( Fix G ⁡ ( L ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions.

A generalization of the Kobayashi–Oshima uniformly bounded multiplicity theorem

Let [Formula: see text] be a minimal parabolic subgroup of a real reductive Lie group [Formula: see text] and [Formula: see text] a closed subgroup of [Formula: see text]. Then it is proved by Kobayashi and Oshima that the regular representation [Formula: see text] contains each irreducible representation of [Formula: see text] at most finitely many times if the number of [Formula: see text]-orbits on [Formula: see text] is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite, where [Formula: see text] are complexifications of [Formula: see text], respectively, and [Formula: see text] is a Borel subgroup of [Formula: see text]. In this paper, we prove that the multiplicities of the representations of [Formula: see text] induced from a parabolic subgroup [Formula: see text] in the regular representation on [Formula: see text] are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic [Formula: see text]-modules.

On the asymptotic enumeration of Cayley graphs

In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

The Koopman Representation and Positive Rokhlin Entropy

In this paper, we study connections between positive entropy phenomena and the Koopman representation for actions of general countable groups. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. We conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley–Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap.

Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

Stationary characters on lattices of semisimple Lie groups

We show that stationary characters on irreducible lattices $\Gamma < G$ Γ < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$ Γ < G , the left regular representation $\lambda _{\Gamma }$ λ Γ is weakly contained in any weakly mixing representation $\pi $ π . We prove that for any such irreducible lattice $\Gamma < G$ Γ < G , any Uniformly Recurrent Subgroup (URS) of $\Gamma $ Γ is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $\Gamma < G$ Γ < G . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

Superfluids in polymer quantum mechanics

In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the non-regular representation of quantum mechanics introduced by the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose–Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau’s criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.

Generalized quasi-regular representation and its applications for shearlet transforms

The construction of continuous shearlet transform has been extended to higher dimensions. It was generalized to a group that is topologically isomorphic to a group of semidirect product of locally compact groups. In this paper, by a unified theoretical linear algebra approach to the representation theory, a class of continuous shearlet transforms obtained from the generalized quasi-regular representation is presented. In order to develop such representation, we utilize a homogeneous space with a relatively invariant Radon measure as tool from computational and abstract harmonic analysis.

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

On the holomorph of finite semisimple groups

Given a finite nonabelian semisimple group 𝐺, we describe those groups that have the same holomorph as 𝐺, that is, those regular subgroups N\simeq G of S(G), the group of permutations on the set 𝐺, such that N_{S(G)}(N)=N_{S(G)}(\rho(G)), where 𝜌 is the right regular representation of 𝐺.