Unitary Representations Of Group Research Articles

A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

Notes on unitary theta representations of compact groups

We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and restriction functors.

Realization of unitary representations of the Lorentz group on de Sitter space

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups G acting on a non-compactly causal symmetric space M=G/H, every irreducible unitary representation of G can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over M. In the present paper we discuss this procedure for the connected Lorentz group G=SO1,d(R)e acting on de Sitter space M=dSd. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain Gℂcut in the complexified group Gℂ=SO1,d(ℂ), which for d=1 specializes to the complex cut plane ℂ∖(−∞,0]. A number of special situations is discussed specifically: (a) The case d=1, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for d>2 is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case d=3, where we obtain rather explicit formulas for the matrix-valued spherical functions.

3d super hyperbolic geometry

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie super group OSpC(1|2) acting on Minkowski (3,1|4)-dimensional super space essentially by Hermitean conjugation. The invariant quadratic form is x1x2−xx¯+ϕψ+ϕ¯ψ¯ in Wigner's coordinates x1,x2∈R and x∈C, where ϕ,ψ are Dirac fermions. The extended action is linear in the super space variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie super group with its Lie algebra. This extension of Wigner's representation opens the door to studying geometry in super hyperbolic three-space with this metric, initiated here with foundations on super geodesics, triangles and tetrahedra, and culminating in a proof of divergence of the volume of a typical ideal tetrahedron and a discussion of the non-zero fermionic correction to the Schläfli formula.

Dissipative dynamics for infinite lattice systems

We study dissipative dynamics constructed by means of non-commutative Dirichlet forms for various lattice systems with multiparticle interactions associated to CCR algebras. We give a number of explicit examples of such models. Using an idea of quasi-invariance of a state, we show how one can construct unitary representations of various groups. Moreover in models with locally conserved quantities associated to an infinite lattice we show that there is no spectral gap and the corresponding dissipative dynamics decay to equilibrium polynomially in time.

Gain total graphs and their spectra via G-phases and group representations

We introduce a definition for the total graph of a gain graph (Γ,ψ) on a group G by using G-phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice of the G-phase associated with ψ. More precisely, we consider the left-right multiplication action of G on the space of all G-phases, proving that its orbits consist of sets of G-phases inducing the same switching equivalence class of gain functions on Γ or, equivalently, on its total graphs. Moreover, we prove that the stabilizers of this action are isomorphic to the centralizers of the sets of gains of closed walks in the associated gain graphs. Our construction is consistent with the existing notions of total graph for signed graphs and, in analogy with the signed case, we are able to explicitly compute the spectrum of the total graph of a regular gain graph over an arbitrary group G, where the spectrum is defined by means of a unitary representation of the gain group G.

Scalar two-point functions at the late-time boundary of de Sitter

We calculate two-point functions of scalar fields of mass m and their conjugate momenta at the late-time boundary of de Sitter with Bunch-Davies boundary conditions, in general d + 1 spacetime dimensions. We perform the calculation using the wavefunction picture and using canonical quantization. With the latter one clearly sees how the late-time field and conjugate momentum operators are linear combinations of the normalized late-time operators αN and βN that correspond to unitary irreducible representations of the de Sitter group with well-defined inner products. The two-point functions resulting from these two different methods are equal and we find that both the autocorrelations of αN and βN and their cross correlations contribute to the late-time field and conjugate momentum two-point functions. This happens both for light scalars m<d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m<\\frac{d}{2}H\\right) $$\\end{document}, corresponding to complementary series representations, and heavy scalars m>d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m>\\frac{d}{2}H\\right) $$\\end{document}, corresponding to principal series representations of the de Sitter group, where H is the Hubble scale of de Sitter. In the special case m = 0, only the βN autocorrelation contributes to the conjugate momentum two-point function in any dimensions and we gather hints that suggest αN to correspond to discrete series representations for this case at d = 3.

Scattering theory with unitary twists

AbstractWe study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillopé and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of 1/2.

Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y(s1,…,sk) with k≥2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k=2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic sp(2k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the su(1,1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y(s1,s2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s1,s2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1,1) is obtained.

Generalized Heisenberg-Weyl groups and Hermite functions

A generalisation of Euclidean and pseudo-Euclidean groups is presented, where the Weyl-Heisenberg groups, well known in quantum mechanics, are involved. A new family of groups is obtained including all the above-mentioned groups as subgroups. Symmetries, like self-similarity and invariance with respect to the orientation of the axes, are properly included in the structure of this new family of groups. Generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of these new groups, are introduced.

Covariant catalysis requires correlations and good quantum reference frames degrade little

Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little.

Anticyclotomic Euler systems for unitary groups

AbstractLet be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature .

Trace formulae for actions of finite unitary groups on cohomology of Artin–Schreier varieties

Associated to a certain additive polynomial, we introduce an Artin–Schreier variety admitting an action of a finite unitary group. We calculate the character of the cohomology as a representation of a finite unitary group. One of our main ingredients is explicit character formulae for Weil representations of unitary groups due to Gérardin. We give another trace formula for a projective hypersurface admitting an action of a finite unitary group.

Geometric wave-front set may not be a singleton

We show that the geometric wave-front set of specific half-integral-depth supercuspidal representations of ramified p p -adic unitary groups is not a singleton.

Decay estimates for matrix coefficients of unitary representations of semisimple Lie groups

Let G be a connected semisimple Lie group with finite centre and K be a maximal compact subgroup thereof. Given a function u on G, we define Au to be the root-mean-square average over K, acting both on the left and the right, of u. We take a positive-real-valued spherical function ϕλ on G, and study the Banach convolution algebra of Cc(G)-functions u with the norm ‖u‖(λ):=∫GAu(x)ϕλ(x)dx. The C⁎ completion of this algebra is an exotic C⁎-algebra on G, in the sense that it lies “between” the reduced C⁎-algebra of G and the full C⁎-algebra of G, and in the sense that it arise as the completion of a star-algebra that does not contain an approximate identity.Using functional analysis and representation theory, we show that for all unitary representations π of G, there exists a unique minimal positive-real-valued spherical function ϕλ on G such that A〈π(⋅)ξ,η〉≤‖ξ‖Hπ‖η‖Hπϕλ. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates |〈π(⋅)ξ,η〉|≤C(ξ,η)ϕλ for K-finite or smooth vectors ξ and η, and it exhibits different decay rates in different directions at infinity in G. Further, if we assume the latter inequality with arbitrary C(ξ,η), we can prove the former inequality and then return to the latter inequality with explicit knowledge of C(ξ,η). Finally, it holds everywhere in G, in contrast to asymptotic estimates which are not global and to Lp estimates which carry no pointwise information.

On Bergman space zero sets. Editors’ comments on the article by Vaughan Jones

This article has two purposes. The first one is to describe what the Editors know on how and when Vaughan Jones worked on the subject of the article published in the same volume. The starting point, in the late 1980’s, was his fascination for a formula giving Murray–von Neumann dimensions of Hilbert spaces of unitary representations of Fuchsian groups. Over the years, he discovered surprising relations of these dimensions with other domains of mathematics. The second purpose is to expose with some details a subject which plays an important role in Jones’ article: the irreducible projective unitary representations of $\operatorname{PSL}\_2(\mathbf{R})$, which constitute a continuous family known as the discrete series, and which have interesting restrictions to various discrete subgroups of $\operatorname{PSL}\_2(\mathbf{R})$.

Quantum walk on orbit spaces

Inspired by the covering-space method in path integrals on multiply connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Lambda}}$ is an arbitrary lattice and $\mathrm{\ensuremath{\Gamma}}$ is a discrete group whose action on $\mathrm{\ensuremath{\Lambda}}$ has no fixed points. We show that the time-evolution kernel on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$ can be written as a weighted sum of time-evolution kernels on $\mathrm{\ensuremath{\Lambda}}$, where the summation is over the orbit of initial point in $\mathrm{\ensuremath{\Lambda}}$ and weight factors are given by a one-dimensional unitary representation of $\mathrm{\ensuremath{\Gamma}}$. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, all of which are constructed in exactly the same way as for the time-evolution kernel.

Geometric construction of Heisenberg–Weil representations for finite unitary groups and Howe correspondences

We give a geometric construction of the Heisenberg–Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for (textrm{Sp}_{2n},textrm{O}_2^-) over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

R⁎-algebras and C⁎-uniqueness of pre-C⁎-algebras

In 1967, Kadison raised the following question, if a self-adjoint operator algebra on a Hilbert space is finitely generated (algebraically) and each self-adjoint operator in it has finite spectrum, is it finite dimensional? In 2021, M. Mori connected the Kadison's question to von Neumann's concept of regular rings given in 1930's, and introduced a notion of R⁎-algebras. In this paper, we study R⁎-algebras, the Kadison's question, and a related question of uniqueness of C⁎-norm on pre-C⁎-algebras. Precisely, we study atomic R⁎-algebras, and find a maximal ultramatricial R⁎-subalgebra of an atomic R⁎-algebra, which properly contains the maximal purely atomic R⁎-subalgebra generated by all finite projections of the algebra. We prove that the set of all ⁎-algebras for which the Kadison's question has an affirmative answer is closed under direct sum, tensor product, and crossed product, but it is not closed under free product. From the aforementioned stable properties of the algebra in the Kadison's question, we find plentiful examples of ⁎-algebras for which the Kadison's question has an affirmative answer, for instance, the crossed product ⁎-algebra B⋊αG, where G is a locally finite group and B is one of the following algebras: an algebra of finite rank operators, an abelian algebra, a finite dimensional algebra, a twisted group algebra satisfying a certain condition, a rotation algebra, or a group measure space construction algebra. We characterize uniqueness of C⁎-norm on a pre-C⁎-algebras in terms of its ⁎-representations, weak containment of its ⁎-representations, ideals of its enveloping C⁎-algebra, or the primitive ideal space of the C⁎-algebra. We also prove that a unital pre-C⁎-algebra has a unique C⁎-norm if and only if, for a faithful ⁎-representation of the algebra, invertibility of a self-adjoint element of the algebra, as an operator on the representation Hilbert space, is independent of the choice of the ⁎-representation. These characterizations provide answers to the question raised by M. Mori in 2021 on finding a sufficient and necessary condition for a ⁎-algebra to have a unique C⁎-norm. For discrete groups, we give a characterization for a group algebra to have a unique C⁎-norm in terms of the group's unitary representations, providing an answer to the question raised by Alekseev in 2019 on characterizing C⁎-unique discrete groups.

Mass dimension one fields with Wigner degeneracy: A theory of dark matter

Whatever dark matter is, it must be one irreducible unitary representation of the inhomogeneous Lorentz group or another. We here develop a formalism of mass dimension one fermions and bosons of spin one half, and show that they provide natural dark matter candidates. By construction, they are covariant under space-time translations and boosts. However, incorporating the rotational symmetry is non-trivial and requires introducing a two-fold Wigner degeneracy thus doubling the degrees of freedom for particles and anti particles from two to four. With Wigner degeneracy, we have a well-defined theory of mass dimension one fields of spin one half that are physically distinct from the Dirac field. They are local, Lorentz covariant and have positive definite Hamiltonians. The developed framework also has the potential to resolve the cosmological constant problem, and supply dark energy.