Finite Gel’fand pairs and their applications to probability and statistics

Ceccherini‐Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo

doi:10.1007/s10958-007-0041-5

Cited by 30 publications

(46 citation statements)

References 43 publications

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“…It admits several different geometric and algebraic presentations. One construction is as follows (see for example [17]). The symmetric group S 2n acts on {1, 2, ..., 2n} and therefore also on the set of partitions of the latter consisting of two-element subsets.…”

Section: Example Of Encodingmentioning

confidence: 99%

Maps, immersions and permutations

Coquereaux

Zuber

2016

J. Knot Theory Ramifications

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We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolourability is no longer automatic and has to be assumed. We thus have two separate countings in non zero genus, that of bicolourable maps and that of general maps. We use a classical method of encoding maps in terms of permutations, on which the constraints of "one-componentness" and of a given genus may be applied. Depending on the orientation issue and on the bicolourability assumption, permutations for a map with n vertices live in S(4n) or in S(2n). In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of S(4n) (resp. S(2n)) under the action of the centralizer of a certain element of S(4n) (resp. S(2n)). This is achieved either by appealing to a formula by Frobenius or by a direct enumeration of these orbits. Applications to knot theory are briefly mentioned.Comment: 46 pages, 18 figures, 9 tables. Version 2: added precisions on the notion used for the equivalence of immersed curves, new references. Version 3: Corrected typos, one array in Appendix B1 was duplicated by mistake, the position of tables and the order of the final sections have been modified, results unchanged. To be published in the Journal of Knot Theory and Its Ramification

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Section: Example Of Encodingmentioning

confidence: 99%

Maps, immersions and permutations

Coquereaux

Zuber

2016

J. Knot Theory Ramifications

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“…We present now some basic elements of the theory of finite Gelfand pairs that will be applied in what follows (see, for example, [CST1], [CST2] and [D] for many applications).…”

Section: ± Some Definitionsmentioning

confidence: 99%

“…The reader is referred to [CST1] for details. A particular example of Gelfand pair is given by the symmetric Gelfand pairs.…”

Section: ± Some Definitionsmentioning

confidence: 99%

A Group of Automorphisms of the Rooted Dyadic Tree and Associated Gelfand Pairs

D'Angeli¹,

Donno²

2009

Rend. Sem. Mat. Univ. Padova

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“…We refer to [9,15,22] for an introduction to Gelfand pairs. Suffice it to mention that finding and understanding Gelfand pairs is an important issue in the representation theory of groups with applications to statistics and probability theory; see for instance [10,14]. Distance transitive actions of groups on the boundary of finite trees were frequently used to construct Gelfand pairs of finite groups; see [12,13,29].…”

Section: Introductionmentioning

confidence: 99%

Groups acting on rooted trees and their representations on the boundary

Kionke

2019

Journal of Algebra

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We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of locally 2-transitive actions. Assuming local 2-transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents.The method can be used to study Gelfand pairs and enables us to answer a question of Grigorchuk. To provide examples, we analyse in detail the local 2-transitivity of GGS-groups. Moreover, our results can be used to determine explicit formulae for zeta functions of induced representations defined by Klopsch and the author.

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Finite Gel’fand pairs and their applications to probability and statistics

Cited by 30 publications

References 43 publications

Maps, immersions and permutations

Maps, immersions and permutations

A Group of Automorphisms of the Rooted Dyadic Tree and Associated Gelfand Pairs

Groups acting on rooted trees and their representations on the boundary

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