Unitary Group Research Articles

Gauging Lie group symmetry in (2+1)d topological phases

We present a general algebraic framework for gauging a 0-form compact, connected Lie group symmetry in (2+1)d topological phases. Starting from a symmetry fractionalization pattern of the Lie group GG, we first extend GG to a larger symmetry group \tilde{G}G̃, such that there is no fractionalization with respect to \tilde{G}G̃ in the topological phase, and the effect of gauging \tilde{G}G̃ is to tensor the original theory with a \tilde{G}G̃ Chern-Simons theory. To restore the desired gauge symmetry, one then has to gauge an appropriate one-form symmetry (or, condensing certain Abelian anyons) to obtain the final result. Studying the consistency of the gauging procedure leads to compatibility conditions between the symmetry fractionalization pattern and the Hall conductance. When the gauging can not be consistently done (i.e. the compatibility conditions can not be satisfied), the symmetry GG with the fractionalization pattern has an ’t Hooft anomaly and we present a general method to determine the (3+1)d topological term for the anomaly. We provide many examples, including projective simple Lie groups and unitary groups to illustrate our approach.

Rigidity of superdense coding

The famous superdense coding protocol of Bennett and Wiesner demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared EPR pair. Our first result is that an arbitrary protocol for achieving this task (where there are no assumptions on the sender’s encoding operations or the dimension of the shared entangled state) is locally equivalent to the canonical Bennett-Wiesner protocol. In other words, the superdense coding task is rigid . In particular, we show that the sender and receiver only use additional entanglement (beyond the EPR pair) as a source of classical randomness. We also investigate several questions about higher-dimensional superdense coding, where the goal is to communicate one of d 2 possible messages by sending a d -dimensional quantum state, for general dimensions d . Unlike the d = 2 case (i.e. sending a single qubit), there can be inequivalent superdense coding protocols for higher d . We present concrete constructions of inequivalent protocols, based on constructions of inequivalent orthogonal unitary bases for all d > 2. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. Our analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest.

Elementary Covering Numbers in Odd-Dimensional Unitary Groups

Elementary Covering Numbers in Odd-Dimensional Unitary Groups

Local-global principle for projective homogeneous spaces of unitary groups

Let K be a complete discrete valued field with residue field k and F the function field of a curve over K. Let A∈2Br(F) be a central simple algebra with an involution σ of any kind and F0=Fσ. Let h be an hermitian space over (A,σ) and G=SU(A,σ,h) if σ is of first kind and G=U(A,σ,h) if σ is of second kind. Suppose that char(k)≠2 and ind(A)≤4. Then we prove that projective homogeneous spaces under G over F0 satisfy a local-global principle for rational points with respect to discrete valuations of F.

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.

Bolstering Policy Support for Disadvantaged Groups Through Humanization

Cooley et al. and Hodson and Doucher show that individuals, individuals within groups, and groups evoke different levels of perceived humanity, and that these differences affect sympathy and willingness to help. In three preregistered experiments, we successfully replicate these findings in a different cultural context (Slovakia). We then test whether manipulating these depictions also affects support for policies that benefit the target. We focus on a disadvantaged ethnic minority (the Roma). Finally, we investigate whether internal (under the beneficiary’s control) versus external attribution (outside of the beneficiary’s control) is a mitigating factor. We confirm individuals and group-compositions evoke higher levels of policy support than groups through increases in perceived humanity. However, this relationship only holds under conditions of external attribution. To humanize disadvantaged groups and bolster policy support, advocates should center their communicative messages around individuals rather than unitary groups and avoid stereotype-enforcing internal attributions.

Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations

This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.

Uniform stability and emergent dynamics of particle and kinetic Lohe matrix models

We study asymptotic behaviors of the second-order Lohe matrix model on the unitary group. For the emergent behaviors, we show that a solution to the Lohe matrix model with identical Hamiltonians always converges to equilibrium using a gradient flow approach. Moreover, we establish the uniform-in-time stability with respect to initial data using the complete aggregation estimate, which can be applied to obtain the corresponding mean-field kinetic equation for the one-oscillator distribution function on the phase space. In fact, we rigorously show that Wasserstein-2 distance between two solutions in different regimes converges to zero in any finite time interval, when the number of oscillators tends to infinity. For the asymptotic emergent dynamics of the kinetic equation, we use an energy estimate and temporal evolution for the diameter to provide sufficient frameworks leading to the complete and practical aggregations for identical and non-identical Hamiltonians, respectively.

Bessel models for real unitary groups: The tempered case

Bessel models for real unitary groups: The tempered case

Twisted forms of classical groups

Twisted forms of classical reductive group schemes are described in a unified way. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, called the augmented odd form algebras, consist of 2 2 -step nilpotent groups with an action of the underlying commutative ring, hence the basic descent theory for them will be developed. Finally, classical isotropic reductive groups are described as odd unitary groups up to an isogeny.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

On the exotic isometry flow of the quadratic Wasserstein space over the real line

Kloeckner discovered that the quadratic Wasserstein space over the real line (denoted by W2(R)) is quite peculiar, as its isometry group contains an exotic isometry flow. His result implies that it can happen that an isometry Φ fixes all Dirac measures, but still, Φ is not the identity of W2(R). This is the only known example of this surprising and counterintuitive phenomenon. Kloeckner also proved that the image of each finitely supported measure under these isometries (and thus under all isometry) is a finitely supported measure. Recently we showed that the exotic isometry flow can be represented as a unitary group on L2((0,1)). In this paper, we calculate the generator of this group, and we show that every exotic isometry (and thus every isometry) maps the set of all absolutely continuous measures belonging to W2(R) onto itself.

헝가리의 반유대주의와 홀로코스트 기억 논쟁

Hungarians' perceptions and debates about the Holocaust show diversified aspects through the socialist period and the period after the transition of the socialist system. Hungarian victim consciousness, or collective perception, which can be called victimhood, lies at the base of Hungarian consciousness, and this consciousness is consumed for political purposes by Hungarian politicians and mass media. By combining the narrative of Hungarian alienation and loneliness with nationalism, the extreme logic advocated by the Hungarian far-right appeared. It is Hungarian nationalism that is used as a nutrient to lead this trend. However, in fact, Hungarian nationalism is also likely to be a fictional phenomenon. Nationalist tendencies in Hungary have grown stronger since the transition in 1989 and are still a highly favored political tool today. For this reason, the Jewish victims are regarded as a 'unitary group' and regarded by nationalist Hungarians as a 'common enemy'. Hungarian right-wing politicians are using this narrative to acquire and maintain power. After all, emphasizing the victimhood of Hungarians using nationalism, which is a very attractive tool for politicians, is a worrisome aspect in that it promotes the revival of the anti-Semitic movement and the fading and contempt of memories of the Holocaust.

The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

Abstract In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p {\operatorname{\mathbb{Q}}_{p}} in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra V, to the setting of infinite dimensional JB-algebras. We prove that the structure group Str(V), the cone preserving group G(Ω) and the automorphism group Aut(V) of the algebra V are embedded Banach-Lie groups of GL(V), and that each of the inclusions Aut(V)⊂G(Ω)⊂Str(V) are of embedded Banach-Lie subgroups. We give a full description of the components of Str(V) via cones, isotopes and central projections. We apply these results to V=B(H)sa the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups Str(V),G(Ω),Aut(V), their Banach-Lie algebras and their connected components. We show that the action of the unitary group of H on Aut(V) has smooth local cross sections, thus Aut(V) is a smooth principal bundle over the unitary group, with structure group S1.

Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach.

In this work ground and excited electronic states of Heisenberg cluster models, in the form of configuration interaction many-body wave functions, are characterized within the spin-adapted Graphical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin are presented, including J1-J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and biomimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reordering, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block-diagonal structure of the Hamiltonian matrix, it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route toward wave functions of reduced multireference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest. In particular, these elements will expand the potential of electronic structure methods in understanding magnetic interactions in exchange-coupled polynuclear transition metal clusters.

Branching rule and coadjoint orbit for Heisenberg Gelfand pairs

Let [Formula: see text] be a closed connected subgroup of the unitary group [Formula: see text] and let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group ([Formula: see text]). We consider the semidirect product [Formula: see text] such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text] and [Formula: see text] be the natural projection. It was pointed out by Lipsman, that the unitary dual [Formula: see text] of [Formula: see text] is in one-to-one correspondence with the space of admissible coadjoint orbits [Formula: see text] (see [10]). Let [Formula: see text] be a generic representation of [Formula: see text] and let [Formula: see text]. To these representations we associate, respectively, the admissible coadjoint orbit [Formula: see text] and [Formula: see text] (via the Lipsman’s correspondence). We denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text] which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity [Formula: see text] of an irreducible [Formula: see text]-module [Formula: see text] occurring in the restriction [Formula: see text] could be read from the coadjoint action of [Formula: see text] on [Formula: see text] In this paper, we show that [Formula: see text] For the special case [Formula: see text] (maximal torus in the unitary group [Formula: see text]) and [Formula: see text], we prove that [Formula: see text]

Simplicity, bounded normal generation, and automatic continuity of groups of unitaries

We show that the commutator subgroup of the group of unitaries connected to the identity in a simple unital C*-algebra is simple modulo its center. We then go on to investigate the role of “regularity properties” in the structure of the special unitary group of a C*-algebra. Under mild assumptions, we show that this group has the invariant automatic continuity property and a unique polish group topology. Strengthening our assumptions in the case of simple C*-algebras, we show that the special unitary group modulo its center has bounded normal generation. These results apply to all simple purely infinite C*-algebras and to all simple nuclear C*-algebras in the “classifiable class”. We show with counterexamples how our conclusions may in general fail if no regularity conditions are imposed on the C*-algebra.

A feedforward unitary equivariant neural network

We devise a new type of feedforward neural network. It is equivariant with respect to the unitary group U(n). The input and output can be vectors in ℂn with arbitrary dimension n. No convolution layer is required in our implementation. We avoid errors due to truncated higher order terms in Fourier-like transformation. The implementation of each layer can be done efficiently using simple calculations. As a proof of concept, we have given empirical results on the prediction of the dynamics of atomic motion to demonstrate the practicality of our approach.

Maps preserving products of commuting elements in von Neumann algebras

A bijective map between specific structures in operator algebras is called a piecewise isomorphism if it preserves products of commuting elements in both directions. We shall show that any bicontinuous piecewise isomorphism between positive cones of invertible elements in von Neumann algebras or between unitary groups of von Neumann algebras can be described in terms of the following parameters: (i) Jordan *-isomorphism between given algebras (ii) one fixed central element in domain (or range) algebra (iii) a hermitian linear map from one algebra into the center of the other one. Especially, in case of factors any piecewise isomorphism between unitary groups is a Jordan *-isomorphism or Jordan *-isomorphism composed with inversion. This extends hitherto known results from von Neumann factors to general von Neumann algebras and brings new Jordan invariants of operator structures.