On the symmetric Gelfand pair (ℋn ×ℋn−1,diag(ℋn−1))

2020 ◽  
pp. 2150085
Author(s):  
Omar Tout
Keyword(s):  
Representation Formula ◽  
Conjugacy Classes ◽  
Gelfand Pair ◽  
Induced Representation ◽  
Hyperoctahedral Group ◽  
Multiplicity Free

We show that the [Formula: see text]-conjugacy classes of [Formula: see text] where [Formula: see text] is the hyperoctahedral group on [Formula: see text] elements, are indexed by marked bipartitions of [Formula: see text] This will lead us to prove that [Formula: see text] is a symmetric Gelfand pair and that the induced representation [Formula: see text] is multiplicity free.

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].

Coadjoint geometry for discretely decomposable restrictions of certain series of representations of indefinite unitary groups

2017 ◽  
Vol 28 (11) ◽  
pp. 1750074
Author(s):  
Salma Nasrin
Keyword(s):  
General Theory ◽  
Geometric Quantization ◽  
Representation Formula ◽  
Unitary Groups ◽  
Unitary Representations ◽  
Semisimple Lie Group ◽  
Derived Functor ◽  
Irreducible Unitary Representations ◽  
Visible Action ◽  
Multiplicity Free

Zuckerman’s derived functor module of a semisimple Lie group [Formula: see text] yields a unitary representation [Formula: see text] which may be regarded as a geometric quantization of an elliptic orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations [Formula: see text] of the indefinite unitary group [Formula: see text] and a family of subgroups [Formula: see text] of [Formula: see text] such that the restriction [Formula: see text] is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of [Formula: see text] with respect to reductive subgroups, II, Ann. of Math. 147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005) 497–549), where [Formula: see text] is not necessarily tempered and [Formula: see text] is not necessarily compact. We prove that the corresponding moment map [Formula: see text] is proper, determine the image [Formula: see text], and compute the Corwin–Greenleaf multiplicity function explicitly.

Trace-free SL(2, ℂ)-representations of Montesinos links

2018 ◽  
Vol 27 (08) ◽  
pp. 1850050 ◽  
Author(s):  
Haimiao Chen
Keyword(s):  
Representation Formula ◽  
Conjugacy Classes

Given a link [Formula: see text], a representation [Formula: see text] is trace-free if the image of each meridian has trace zero. We determine the conjugacy classes of trace-free representations when [Formula: see text] is a Montesinos link.

scholarly journals The center of the wreath product of symmetric group algebras

2021 ◽  
Vol 31 (2) ◽  
pp. 302-322
Author(s):  
O. Tout ◽  
Keyword(s):  
Symmetric Group ◽  
Recent Result ◽  
Wreath Product ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Hyperoctahedral Group ◽  
Symmetric Group Algebras

We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.

Square-Integrability of Induced Representations of Semidirect Products

1998 ◽  
Vol 10 (03) ◽  
pp. 301-313 ◽  
Author(s):  
P. Aniello ◽  
Gianni Cassinelli ◽  
Ernesto de Vito ◽  
Alberto Levrero
Keyword(s):  
Representation Formula ◽  
Irreducible Unitary Representation ◽  
Dual Group ◽  
Unitary Representations ◽  
Induced Representation ◽  
Semidirect Products ◽  
Induced Representations ◽  
Stability Group ◽  
The Stability ◽  
Square Integrability

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form [Formula: see text] where x0 is in the dual group of A, G0 is the stability group of x0 and m is an irreducible unitary representation of G0∩H. We give a new selfcontained proof of the following result: the induced representation [Formula: see text] is square-integrable if and only if the orbit G[x0] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of [Formula: see text].

scholarly journals Groups with boundedly finite conjugacy classes of commutators

2018 ◽  
Vol 69 (3) ◽  
pp. 1047-1051 ◽  
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky
Keyword(s):  
Conjugacy Classes ◽  
Finite Conjugacy

scholarly journals Connected perimeter of planar sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Dayrens ◽  
Simon Masnou ◽  
Matteo Novaga ◽  
Marco Pozzetta
Keyword(s):  
Minimization Problem ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Representation Formula ◽  
Steiner Trees ◽  
Lower Semicontinuous Envelope ◽  
Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

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