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

Abstract

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.

Induced Representations of the Infinite Symmetric Group

“…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.…”

“…As shown in [22], in this case one can obtain a complete spectral analysis of the induced representations. We present these details in Sec.…”

“…Theorem 5 is proved there using the so-called tensor model of two-row representations of the symmetric groups (i.e., representations induced from two-block Young subgroups). See [22] for a detailed description of this model (in particular, explicit formulas for the Gelfand-Tsetlin basis). A partition N = A ∪ B of type ∞ 2 is uniquely determined by an infinite sequence ξ = ξ 1 ξ 2 .…”

“…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.…”

“…In this paper, we consider the first problem for some classes of induced representations. The first result in this direction, obtained in [22] and specified below, is that the representations induced from Young subgroups corresponding to partitions into two infinite blocks are simple and irreducible, and their spectral measures are Markov measures. Another nontrivial example considered in this paper is the class of induced representations corresponding to partitions with infinitely many singletons and finitely many finite blocks of length greater than one.…”