History and variation on the theme of the frobenius reciprocity theorem

1991 ◽  
Vol 13 (3) ◽  
pp. 68-71 ◽  
Author(s):  
Floyd L. Williams
Keyword(s):  
Reciprocity Theorem ◽  
Frobenius Reciprocity

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

2007 ◽  
Vol 18 (02) ◽  
pp. 137-164 ◽  
Author(s):  
CLAUDIA PINZARI
Keyword(s):  
Quantum Groups ◽  
Reciprocity Theorem ◽  
Sufficient Condition ◽  
Alternative Definition ◽  
Topological Transitivity ◽  
Frobenius Reciprocity ◽  
Induced Representations ◽  
C Algebras ◽  
Quotient Spaces ◽  
Compact Quantum Groups

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.

The Explicit Fourier Decomposition of L2SO(n)/SO(n - m))

1975 ◽  
Vol 27 (2) ◽  
pp. 294-310 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Spherical Harmonics ◽  
Reciprocity Theorem ◽  
Irreducible Representations ◽  
Frobenius Reciprocity ◽  
Fourier Decomposition ◽  
Direct Sum ◽  
New Phenomena

The decomposition of L2SO(n)/SO(n - m)) into a direct sum of irreducible representations of SO(n) is given abstractly by the branching theorem and the Frobenius reciprocity theorem [1]. The goal of this paper is to obtain this decomposition explicitly, generalizing the theory of spherical harmonics (m = 1). The case m = 2 has been studied in Levine [6], and the case 2m ≦ n in Gelbart [3]. Our results shed more light on these cases as well as revealing new phenomena which only occur when 2m > n.

Restricting and Inducing on Inner Products of Representations of Finite Groups

1975 ◽  
Vol 27 (6) ◽  
pp. 1349-1354
Author(s):  
G. de B. Robinson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Reciprocity Theorem ◽  
Frobenius Reciprocity ◽  
Inner Products ◽  
Starting Point ◽  
Representations Of Finite Groups ◽  
A Finite Group

Of recent years the author has been interested in developing a representation theory of the algebra of representations [5; 6] of a finite group G, and dually of its classes [7]. In this paper Frobenius’ Reciprocity Theorem provides a starting point for the introduction of the inverses R-1 and I-1 of the restricting and inducing operators R and I. The condition under which such inverse operations are available is that the classes of G do not splitin the subgroup Ĝ. When this condition is satisfied the application of these operations to inner products is of interest.

A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

2019 ◽  
Vol 100 (2) ◽  
pp. 317-322
Author(s):  
H. KUMUDINI DHARMADASA ◽  
WILLIAM MORAN
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Banach Spaces ◽  
Closed Subgroup ◽  
General Setting ◽  
Locally Compact Group ◽  
Reciprocity Theorem ◽  
Representation Space ◽  
Locally Compact ◽  
Frobenius Reciprocity

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.

The Dual of Frobenius' Reciprocity Theorem

1973 ◽  
Vol 25 (5) ◽  
pp. 1051-1059 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Finite Group ◽  
Reciprocity Theorem ◽  
Integral Representations ◽  
Irreducible Representations ◽  
Frobenius Reciprocity ◽  
Multiplication Tables ◽  
Symmetry Properties ◽  
Structure Constants ◽  
A Finite Group

In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.

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

1993 ◽  
Vol 16 (1) ◽  
pp. 61-66
Author(s):  
Robert A. Bekes ◽  
Peter J. Hilton
Keyword(s):  
Reciprocity Theorem ◽  
Point Of View ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Adjoint Functors ◽  
Locally Compact ◽  
Frobenius Reciprocity ◽  
Theoretic Point

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

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