Unitary Representations Of Group Research Articles

The density theorem for projective representations via twisted group von Neumann algebras

We consider converses to the density theorem for square-integrable, irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that the restriction of such a σ-projective unitary representation π of a unimodular, second-countable group G to a lattice Γ extends to a Hilbert module over the twisted group von Neumann algebra of (Γ,σ). We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that the center-valued von Neumann dimension reduces to the scalar value dπvol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to π and Γ. In particular, we characterize when a lattice in the time-frequency plane of a second-countable, locally compact abelian group admits a Gabor frame or Gabor Riesz sequence.

On capacity of quantum channels generated by irreducible projective unitary representations of finite groups

We study mixed unitary quantum channels generated by irreducible projective unitary representations of finite groups. Under some assumptions on the probability distribution determining a mixture the classical capacity of the channel is found. Examples illustrating the techniques are given.

Big principal series, -adic families and -invariants

In earlier work, the first named author generalized the construction of Darmon-style$\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic$\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in$p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic$\mathcal {L}$-invariants of Hilbert modular forms of parallel weight$2$are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur$\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur$\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

Complexity for Conformal Field Theories in General Dimensions.

We study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti-de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states.

A question of Katz and Tiep on representations of finite general unitary groups

Katz and Tiep proved a characterization of the total Weil character of finite special unitary groups for any integer and any prime power q other than in terms of the character values and irreducible constituents. They asked whether a similar characterization can be done for finite general unitary groups In this article, we prove that such characterization for can be done up to endomorphisms and tensoring by real linear characters.

Theta lifting for tempered representations of real unitary groups

We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.

Reflections on Virasoro circuit complexity and Berry phase

Recently, the notion of circuit complexity defined in symmetry group manifolds has been related to geometric actions which generally arise in the coadjoint orbit method in representation theory and play an important role in geometric quantization. On the other hand, it is known that there exists a precise relation between geometric actions and Berry phases defined in group representations. Motivated by these connections, we elaborate on a relation between circuit complexity and the group theoretic Berry phase. As the simplest setup relevant for holography, we discuss the case of two dimensional conformal field theories. In the large central charge limit, we identify the computational cost function with the Berry connection in the unitary representation of the Virasoro group. We then use the latter identification to express the Berry phase in terms of the Virasoro circuit complexity. The former can be seen as the holonomy of the Berry connection along the path in the group manifold which defines the protocol. In addition, we derive a proportionality relation between Virasoro circuit complexity and the logarithm of the inner product between a particularly chosen reference state and the prepared target state. In this sense, the logarithmic formula turns out to be approximating the complexity up to some additive constant if the building blocks of the circuit are taken to be the underlying symmetry gates. Predictions based on this formula have recently been shown to coincide with the holographic complexity proposals and the path integral optimization procedure. The found connections may therefore help to better understand such coincidences. We also discuss that our findings, put together with earlier observations, may suggest a connection between the Virasoro Berry phase and the complexity measure in the path integral optimization proposal.

On a family of unitary representations of mapping class groups

For a compact surface S = S_{g,n} with 3g + n \geq 4 , we introduce a family of unitary representations of the mapping class group \operatorname{Mod}(S) based on the space of measured foliations. or this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of \operatorname{Mod}(S) on the Teichmüller space of S . Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups.

Free group representations: Duplicity and oddity on the boundary

We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.

Path Integrals in a Multiply-Connected Configuration Space (50 Years After)

The proposal made 50 years ago by Schulman (1968), Laidlaw & Morette-DeWitt (1971) and Dowker (1972) to decompose the propagator according to the homotopy classes of paths was a major breakthrough: it showed how Feynman functional integrals opened a direct window on quantum properties of topological origin in the configuration space. This paper casts a critical look at the arguments brought by this series of papers and its numerous followers in an attempt to clarify the reason why the emergence of the unitary linear representation of the first homotopy group is not only sufficient but also necessary.

Manifold topology, observables, and gauge group

The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with positions and displacements on the manifold. The guiding, physically motivated, principles are (i) locality, i.e., the generating role of the algebras localized in small, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, “topological,” subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering M̃ of M and on the decomposition of L2(M̃) according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the permutation group in particle statistics are discussed.

A contraction of the principal series representations of $SL(2,\mathbf{R})$

We introduce and study a contraction of the principal series representations of $SL(2,\mathbf{R})$ to the unitary irreducible representations of the Heisenberg group. We interpret the contraction results in terms of Weyl correspondences on the coadjoint orbits associated with the representations.

Square root p-adic L-functions, I : Construction of a one-variable measure

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1/2,BC(\pi) \times BC(\pi'))$, where $\pi$ and $\pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n) \times GL(n-1)$. This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $\pi$ is a holomorphic representation and $\pi'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $\pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.

Local Langlands correspondence for unitary groups via theta lifts

Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which is due to Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this paper provides an alternative approach to the works of Kaletha-Mínguez-Shin-White and Mœglin-Renard.

Solving Rubik’s cube via quantum mechanics and deep reinforcement learning

Rubik’s cube is one of the most famous combinatorial puzzles involving nearly 4.3 × 1019 possible configurations. Its mathematical description is expressed by the Rubik’s group, whose elements define how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the cube from its geometrical constraints. Cubies are described by single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the cube, as a geometrical object, shows symmetries which are broken when driven away from this configuration. For each of such symmetries, we build a Hamiltonian operator. When a Hamiltonian lies in its ground state, the respective symmetry of the cube is preserved. When all such symmetries are preserved, the configuration of the cube matches the solution of the game. To reach the ground state of all the Hamiltonian operators, we make use of a deep reinforcement learning algorithm based on a Hamiltonian reward. The cube is solved in four phases, all based on a respective Hamiltonian reward based on its spectrum, inspired by the Ising model. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.

Inductive limits of compact quantum groups and their unitary representations

We introduce inductive limits of compact quantum groups and their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantization of characters and central probability measures in the asymptotic representation theory. We also study tensor product representations of the infinite-dimensional quantum unitary group. As a by-product, we give an explicit representation-theoretic interpretation to certain transformations that play an important role in analyzing q-central probability measures on the paths in the q-Gelfand–Tsetlin graph.

Gauging the Higher-Spin-Like Symmetries by the Moyal Product. II

Continuing the study of the Moyal Higher Spin Yang–Mills theory started in our previous paper we provide a detailed discussion of matter coupling and the corresponding tree-level amplitudes. We also start the investigation of the spectrum by expanding the master fields in terms of ordinary spacetime fields. We note that the spectrum can be consistent with unitarity while still preserving Lorentz covariance, albeit not in the usual way, but by employing an infinite-dimensional unitary representation of the Lorentz group.

On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Consider an infinite homogeneous tree \mathcal{T}_n of valence n+1 , its group Aut (\mathcal{T}_n) of automorphisms, and the group Hier (\mathcal{T}_n) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of \mathcal{T}_n that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut (\mathcal{T}_n) is spherical in Hier (\mathcal{T}_n) , i.e., any irreducible unitary representation of Hier (\mathcal{T}_n) contains at most one Aut (\mathcal{T}_n) -fixed vector. We present a combinatorial description of the space of double cosets of Hier (\mathcal{T}_n) with respect to Aut (\mathcal{T}_n) and construct a “new” family of spherical representations of Hier (\mathcal{T}_n) . We also show that the Thompson group Th has PSL (2,\mathbb{Z}) -spherical unitary representations.

Automorphic ℒ‐invariants for reductive groups

Abstract Let G be a reductive group over a number field F, which is split at a finite place 𝔭 {\mathfrak{p}} of F, and let π be a cuspidal automorphic representation of G, which is cohomological with respect to the trivial coefficient system and Steinberg at 𝔭 {\mathfrak{p}} . We use the cohomology of 𝔭 {\mathfrak{p}} -arithmetic subgroups of G to attach automorphic ℒ {\mathcal{L}} -invariants to π. This generalizes a construction of Darmon (respectively Spieß), who considered the case G = GL 2 {G={\mathrm{GL}}_{2}} over the rationals (respectively over a totally real number field). These ℒ {\mathcal{L}} -invariants depend a priori on a choice of degree of cohomology, in which the representation π occurs. We show that they are independent of this choice provided that the π-isotypic part of cohomology is cyclic over Venkatesh’s derived Hecke algebra. Further, we show that automorphic ℒ {\mathcal{L}} -invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic ℒ {\mathcal{L}} -invariants are equal to the Fontaine–Mazur ℒ {\mathcal{L}} -invariants of the associated Galois representation.

On unitary representations of algebraic groups over local fields

Let G \mathbf {G} be an algebraic group over a local field k \mathbf {k} of characteristic zero. We show that the locally compact group G ( k ) \mathbf {G}(\mathbf {k}) consisting of the k \mathbf {k} -rational points of G \mathbf {G} is of type I. Moreover, we complete Lipsman’s characterization of the groups G \mathbf {G} for which every irreducible unitary representation of G ( k ) \mathbf {G}(\mathbf {k}) is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.