Frobenius Reciprocity Theorem Research Articles

From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n-1)⊂S(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(n-1)\\subset S(n)$$\\end{document}. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.

Induced Representation Method in the Theory of Electron Structure and Superconductivity

It is shown that the application of theorems of induced representations method, namely, Frobenius reciprocity theorem, transitivity of induction theorem, and Mackey theorem on symmetrized squares, makes simplifying standard techniques in the theory of electron structure and constructing Cooper pair wavefunctions on the basis of one-electron solid-state wavefunctions possible. It is proved that the nodal structure of topological superconductors in the case of multidimensional irreducible representations is defined by additional quantum numbers. The technique is extended on projective representations in the case of nonsymmorphic space groups and examples of applications for topological superconductors UPt3 and Sr2RuO4 are considered.

A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

Frobenius reciprocity and the Haagerup tensor product

In the context of operator-space modules over C ∗ C^* -algebras, we give a complete characterisation of those C ∗ C^* -correspondences whose associated Haagerup tensor product functors admit left adjoints. The characterisation, which builds on previous joint work with N. Higson, exhibits a close connection between the notions of adjoint operators and adjoint functors. As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H H of a locally compact group G G admits a left adjoint in this setting if and only if H H is cocompact in G G . The adjoint functor is given by the Haagerup tensor product with the operator-theoretic adjoint of Rieffel’s induction bimodule.

EMBEDDING ERGODIC ACTIONS OF COMPACT QUANTUM GROUPS ON C*-ALGEBRAS INTO QUOTIENT SPACES

The notion of compact quantum subgroup is revisited and an alternative definition is given. Induced representations are considered and a Frobenius reciprocity theorem is obtained. A relationship between ergodic actions of compact quantum groups on C*-algebras and topological transitivity is investigated. A sufficient condition for embedding such actions in quantum quotient spaces is obtained.

Square-integrability modulo a subgroup

We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of square-integrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.

Spectra of the Laplacian on the Cayley projective plane

Let M = G/K be a compact homogeneous space of a compact semi-simple Lie group G. Let V be a complex homogeneous vector bundle on M. The group G acts naturally on the space of sections T{V) of V. By a theorem of Peter and Weyl, T(V) is a unitary direct sum of finitedimensional representations of G. It is an important problem to decompose T{V) into irreducible (j-modules. By the Frobenius reciprocity theorem, the problem is divided into two parts:

Mackey Theorem and Two-Electron Wave Function of a Multi-Centre System

The two-electron wave function in a system of many equivalent atoms is investigated group-theoretically. It is shown that the classification of different types of two-electron (two-hole) localizations can be made by the double-coset decomposition of the symmetry group with respect to the local subgroup, and that the group appearing in the Mackey theorem can be used for the additional classification of states. The Mackey theorem on symmetrized squares and the generalized Frobenius reciprocity theorem are applied to the construction of two-electron states in octahedral symmetry.

Induced representation and Frobenius reciprocity for compact quantum groups

Unitary representations of compact quantum groups have been described as isometric comodules. The notion of an induced representation for compact quantum groups has been introduced and an analogue of the Frobenius reciprocity theorem is established.

Modified symmetry generators and the geometric phase

Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately Sd (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.

How to obtain easily the induced representations of point groups: the icosahedral point groups

With tables of subduced representations as a starting point and use of the Frobenius reciprocity theorem, a simple method to obtain induced representations is given. Tables are given of the 22 induced representations of 532 (I) and of the 84 induced representations of {\overline 5}{\overline 3}2/m (Ih).

Induction and restriction as adjoint functors on representations of locally compact groups

In this paper the Frobenius Reciprocity Theorem for locally compact groups is looked at from a category theoretic point of view.

History and variation on the theme of the frobenius reciprocity theorem

The German mathematician Georg Ferdinand Frobenius (1849-1917) was born in Berlin on 26 October 1849, the son of Christian Ferdinand Frobenius. After at tending the Joachimthal Gymnasium, Frobenius began his mathematical studies at Gottingen in 1867. Under the direction of the inimitable Karl Weierstrass he earned a doctorate three years later in Berlin. In ensuing years, Frobenius made fundamental discoveries in such diverse areas as matrix theory mod p and 0-functions in several variables. His particularly productive work during the 1870-75 period is chronicled in Crelle's Journal. The major achievements of Frobenius are undoubtedly in the theory of group representat ions--a theory he initiated in 1896 at the age of 47. Along with its three historic roots--Galois theory, Lie theory, and number theory-i t stands as a cornerstone of modern mathematics. group, so we begin with the latter notion. It appears, implicitly, as early as 1801 in the work of Gauss--viz . , in his Dzsquisitiones Arithmetzcae--and m later years, beginning in 1878, Richard Dedekind also considered the notion (again implicitly). An extensive correspondence be tween Dedekind and Frobenius, spanning periods from 1882-83 and from 1895-98, provides valuable insight into the development of Frobenius's creation. A fascinating and learned account of this cor-

N-Fold tensor products of GL( N, [formula omitted]) and decomposition of fock spaces

Fock spaces over n × N complex matrices are introduced to deal with the reduction of n-fold tensor products of representations of GL( N, C ). These spaces are decomposed into orthogonal direct sums of finite-dimensional polynomial spaces P ( M) that are invariant under GL( N, C ). A Frobenius reciprocity theorem is proved which relates the number of times an irreducible representation of GL( N, C ) appears in P ( M) to the number of times this irreducible representation contains a representation of its diagonal subgroup. A GL( n, C ) action is shown to commute with the GL( N, C ) action and is used to find maps that send irreducible representations of GL( N, C ) into the P ( M) spaces. Invariant Casimir operators are introduced to deal with the multiplicity occurring in n-fold tensor products.

Characters of Finite Quasigroups II: Induced Characters

Induced characters for finite quasigroups are defined, simplifying and generalizing the usual definition for groups. The Frobenius Reciprocity Theorem and an analogue of Artin's Theorem for these characters are proved. Character rings for quasigroups are examined. Induced characters are then used to build the character table of the octonion loop.

On a Frobenius reciprocity theorem for locally compact groups II

On a Frobenius reciprocity theorem for locally compact groups II

On a Frobenius reciprocity theorem for locally compact groups

On a Frobenius reciprocity theorem for locally compact groups

Dynamical symmetry breaking and energy transfer pathways in local modes of benzene

Local modes are discussed as a manifestation of dynamical symmetry breaking. For benzene C-H fundamentals, the symmetry is lowered from D/sub 6h/ to C/sub 2//sub v/ when the excitation process singles out one of the local modes. The paradox of symmetry breaking is removed, however, when one takes into account dynamical tunneling mechanisms which restore D/sub 6h/ symmetry through energy transfer among the local modes. The situation is related to permutation-inversion isomerism in molecules with identical nuclei, with restoration of full permutation-inversion symmetry by means of tunneling among the isomers. It is shown that mathematically, the analogy subsists in the treatment of both cases in terms of induced and subduced representations and the Frobenius reciprocity theorem. It is then shown how local modes symmetry breaking may be incorporated within the framework of fundamental permutation-inversion symmetry. Splittings in local modes spectra are parameterized phenomenologically by classifying the dynamical tunneling mechanisms, or energy transfer pathways among local modes, available to the system. The phenomenological analysis of observed spectra yields relative strengths for the various pathways. The results are compared with predictions of a recent model and with intuition.

Spherical Distributions on Lie Groups and C ∞ Vectors

Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation 8 of K a class of distributions analogous to the class of spherical functions of height 8 familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel's theorem to arbitrary induced representations of Lie

Spherical distributions on Lie groups and 𝐶^{∞} vectors

Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation δ \delta of K a class of distributions analogous to the class of spherical functions of height δ \delta familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel’s theorem to arbitrary induced representations of Lie groups.