On different models of representations of the infinite symmetric group

Цилевич, Н. В.; Vershik, A. M.

doi:10.1016/j.aam.2005.09.007

Cited by 15 publications

(7 citation statements)

References 7 publications

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“…In this section, we describe the so-called tensor model of two-row representations of the symmetric groups, which was suggested by the first author and studied in [4] (see also [5]).…”

Section: The Tensor Model Of Two-row Representationsmentioning

confidence: 99%

Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

Vershik¹,

Цилевич²

2007

Theory Probab. Appl.

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We show that the class of so-called Markov representations of the infinite symmetric group S N , associated with Markov measures on the space of infinite Young tableaux, coincides with the class of simple representations, i.e., inductive limits of representations with simple spectrum. The spectral measure of an arbitrary representation of S N with simple spectrum is equivalent to a multi-Markov measure on the space of Young tableaux. We also show that the representations of S N induced from the identity representations of two-block Young subgroups are Markov and find explicit formulas for the transition probabilities of the corresponding Markov measures. The induced representations are studied with the help of the tensor model of two-row representations of the symmetric groups; in particular, we deduce explicit formulas for the Gelfand-Tsetlin basis in the tensor models.We show that the class of so-called Markov representations of the infinite symmetric group S N , associated with Markov measures on the space of infinite Young tableaux, coincides with the class of simple representations, i.e., inductive limits of representations with simple spectrum. The spectral measure of an arbitrary representation of S N with simple spectrum is equivalent to a multi-Markov measure on the space of Young tableaux. We also show that the representations of S N induced from the identity representations of two-block Young subgroups are Markov and find explicit formulas for the transition probabilities of the corresponding Markov measures. The induced representations are studied with the help of the tensor model of two-row representations of the symmetric groups; in particular, we deduce explicit formulas for the Gelfand-Tsetlin basis in the tensor models.

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“…In this section, we describe the so-called tensor model of two-row representations of the symmetric groups, which was suggested by the first author and studied in [4] (see also [5]).…”

Section: The Tensor Model Of Two-row Representationsmentioning

confidence: 99%

Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

Vershik¹,

Цилевич²

2007

Theory Probab. Appl.

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“…Combinatorial encoding can also be applied to systems with a finite or countable set of states. Note that a related isomorphism was considered in [24] from the viewpoint of representation theory (the so-called concomitant representations). In terms of the RSK algorithm, this case corresponds to discrete central measures and is much simpler than the case of the Plancherel measure.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial encoding of Bernoulli schemes and the asymptotic behavior of Young tableaux

Vershik¹

2020

Preprint

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We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson-Schensted-Knuth) correspondence. In the first case, the decodability is a quite simple fact, while in the second case, this is a nontrivial result obtained by D. Romik and P. Śniady and based on the papers [2], [12], and others. We comment on the proofs from the viewpoint of the theory of measurable partitions; another proof, using representation theory and generalized Schur-Weyl duality, will be presented elsewhere. We also study a new dynamics of Bernoulli variables on P -tableaux and find the limit 3D-shape of these tableaux.

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“…It is well known (see, e.g., [10,14,22] and also Sec. 5.2) that the representation I n can be realized in the space of symmetric tensors of rank k and dimension n. Then the representation I (∞,k) can be realized in the space of infinite-dimensional symmetric tensors of rank k.…”

Section: Proposition 1 the Representation Of The Infinite Symmetric mentioning

confidence: 99%

“…We consider these two problems as the main problems of this theory, and its main analytic and probabilistic component is the analysis of spectral measures; see, e.g., the authors' papers [14,22,21]. In this paper, we consider the first problem for some classes of induced representations.…”

Section: Introductionmentioning

confidence: 99%

Induced representations of the infinite symmetric group and their spectral theory

Vershik

Цилевич

2007

Dokl. Math.

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Abstract:We study the representations of the infinite symmetric group induced from the identity representations of Young subgroups. It turns out that such induced representations can be either of type I or of type II. Each Young subgroup corresponds to a partition of the set of positive integers; depending on the sizes of blocks of this partition, we divide Young subgroups into two classes: large and small subgroups. The first class gives representations of type I, in particular, irreducible representations. The most part of Young subgroups of the second class give representations of type II and, in particular, von Neumann factors of type II. We present a number of various examples. The main problem is to find the so-called spectral measure of the induced representation. The complete solution of this problem is given for two-block Young subgroups and subgroups with infinitely many singletons and finitely many finite blocks of length greater than one.

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On different models of representations of the infinite symmetric group

Cited by 15 publications

References 7 publications

Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

Combinatorial encoding of Bernoulli schemes and the asymptotic behavior of Young tableaux

Induced representations of the infinite symmetric group and their spectral theory

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