Unitary representations of infinite wreath products

Abstract Surface operators are among the most important observables of the 6d $$ \mathcal{N} $$ N = (2, 0) theory. Here we apply the tools of defect CFT to study local operator insertions into the 1/2-BPS plane. We first relate the 2-point function of the displacement operator to the expectation value of the bulk stress tensor and translate this relation into a constraint on the anomaly coefficients associated with the defect. Secondly, we study the defect operator expansion of the stress tensor multiplet and identify several new operators of the defect CFT. Technical results derived along the way include the explicit supersymmetry tranformations of the stress tensor multiplet and the classification of unitary representations of the superconformal algebra preserved by the defect.

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Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.9 ◽

2021 ◽

pp. 1-36

Author(s):

ARIE LEVIT ◽

ALEXANDER LUBOTZKY

Keyword(s):

Ergodic Theorem ◽

Abelian Groups ◽

Wreath Products ◽

Finitely Generated ◽

Metabelian Groups ◽

Amenable Groups ◽

Pointwise Ergodic Theorem ◽

Lamplighter Group ◽

Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

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RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000165 ◽

2021 ◽

pp. 1-37

Author(s):

Martijn Caspers

Keyword(s):

Quantum Group ◽

Quantum Groups ◽

Riesz Transform ◽

Wreath Products ◽

Compact Quantum Group ◽

Riesz Transforms ◽

Markov Semigroup ◽

Markov Semigroups ◽

Free Products ◽

Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

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On the Dowling and Rhodes lattices and wreath products

Journal of Algebra ◽

10.1016/j.jalgebra.2021.02.018 ◽

2021 ◽

Author(s):

Stuart Margolis ◽

John Rhodes ◽

Pedro V. Silva

Keyword(s):

Wreath Products

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Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽

10.3390/sym13061060 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1060

Author(s):

Enrico Celeghini ◽

Manuel Gadella ◽

Mariano A. del del Olmo

Keyword(s):

Hilbert Space ◽

Fourier Transform ◽

Heisenberg Group ◽

Real Line ◽

Hermite Functions ◽

Unitary Representations ◽

Weyl Groups ◽

The Real ◽

Symmetry Properties ◽

The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

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