scholarly journals Clebsch–Gordan coefficients of discrete groups in subgroup bases

2018 ◽  
Vol 33 (10) ◽  
pp. 1850055
Author(s):  
Gaoli Chen
Keyword(s):  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Conjugate ◽  
Discrete Groups

We express each Clebsch–Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group [Formula: see text] in terms of embedding factors of its subgroups [Formula: see text] and [Formula: see text].

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

scholarly journals Quasi-flat representations of uniform groups and quantum groups

2019 ◽  
Vol 18 (08) ◽  
pp. 1950155 ◽  
Author(s):  
Teodor Banica ◽  
Alexandru Chirvasitu
Keyword(s):  
Quantum Groups ◽  
Discrete Group ◽  
Homogeneous Spaces ◽  
Model Space ◽  
Representation Formula ◽  
Discrete Groups ◽  
Universal Model ◽  
Induced Representation ◽  
Negative Results ◽  
Number Formula

Given a discrete group [Formula: see text] and a number [Formula: see text], a unitary representation [Formula: see text] is called quasi-flat when the eigenvalues of each [Formula: see text] are uniformly distributed among the [Formula: see text]th roots of unity. The quasi-flat representations of [Formula: see text] form altogether a parametric matrix model [Formula: see text]. We compute here the universal model space [Formula: see text] for various classes of discrete groups, notably with results in the case where [Formula: see text] is metabelian. We are particularly interested in the case where [Formula: see text] is a union of compact homogeneous spaces, and where the induced representation [Formula: see text] is stationary in the sense that it commutes with the Haar functionals. We present several positive and negative results on this subject. We also discuss similar questions for the discrete quantum groups, proving a stationarity result for the discrete dual of the twisted orthogonal group [Formula: see text].

scholarly journals Induced Coactions of Discrete Groups on C*-Algebras

1999 ◽  
Vol 51 (4) ◽  
pp. 745-770 ◽  
Author(s):  
Siegfried Echterhoff ◽  
John Quigg
Keyword(s):  
Quotient Group ◽  
Discrete Group ◽  
Discrete Groups ◽  
Crossed Products ◽  
C Algebras ◽  
Morita Equivalent ◽  
Close Relationship

AbstractUsing the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) of G. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products Ind D ×IndδG and D ×δG/N are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

Groups with Representations of Bounded Degree

1964 ◽  
Vol 16 ◽  
pp. 299-309 ◽  
Author(s):  
I. M. Isaacs ◽  
D. S. Passman
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Bounded Degree ◽  
Normal Abelian Subgroup

Let G be a discrete group with group algebra C[G] over the complex numbers C. In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n, then all irreducible representations of C[G] have degree ≤n. Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C[G] have degree ≤n, then G has an abelian subgroup of index not greater than some function of n. (The degree of a representation of C[G] for arbitrary G is defined precisely in § 3.)

scholarly journals Boundaries of reduced C * C^{*} -algebras of discrete groups

2017 ◽  
Vol 2017 (727) ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthew Kennedy
Keyword(s):  
Open Problem ◽  
Discrete Group ◽  
Discrete Groups ◽  
Algebraic Construction ◽  
C Algebras

AbstractFor a discrete groupThis operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove thatThe algebraIt is a longstanding open problem to determine which groups are

scholarly journals Amenable actions of discrete groups

1993 ◽  
Vol 13 (2) ◽  
pp. 289-318 ◽  
Author(s):  
G. A. Elliott ◽  
T. Giordano
Keyword(s):  
Discrete Group ◽  
Structure Theorem ◽  
Discrete Groups ◽  
Countable Discrete Group

AbstractA structure theorem is established for amenable actions of a countable discrete group.

scholarly journals Examples of Calabi-Yau threefolds with small Hodge numbers

2021 ◽  
Vol 2081 (1) ◽  
pp. 012030
Author(s):  
A O Shishanin
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Gauge Group ◽  
Complete Intersection ◽  
Euler Characteristic ◽  
Discrete Group ◽  
Free Action ◽  
Discrete Groups ◽  
Hodge Numbers ◽  
Fixed Point Sets

Abstract We observe some suitable examples of Calabi-Yau threefolds for heterotic superstring compactifications. It is reasonable to seek CY threefolds with Euler characteristic equals ±6 because of generation’s number. Hosotani mechanism for violations of the gauge group by the Wilson loops requires such CY space has a non-trivial fundamental group. These spaces can be obtained by factoring the complete intersection Calabi-Yau spaces by the free action of some discrete group. Also we shortly discuss cases when discrete groups act with fixed point sets.

scholarly journals Positive definite functions and cut-off for discrete groups

2020 ◽  
pp. 1-17
Author(s):  
Amaury Freslon
Keyword(s):  
Random Walks ◽  
Quantum Groups ◽  
Discrete Group ◽  
Coxeter Groups ◽  
Free Groups ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Definite Function ◽  
Discrete Groups ◽  
Compact Quantum Groups

Abstract We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

Disjoint Hypercyclicity and Weighted Translations on Discrete Groups

2017 ◽  
Vol 60 (4) ◽  
pp. 712-720
Author(s):  
Chung-Chuan Chen
Keyword(s):  
Lebesgue Space ◽  
Discrete Group ◽  
Necessary Condition ◽  
Discrete Groups ◽  
Translation Operators ◽  
Weighted Translation ◽  
Sufficient And Necessary Condition ◽  
Disjoint Hypercyclicity

Let 1 ≤ p < ∞, and let G be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space ℓp(G) to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.

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