A Realization of the Irreducible Representations of S n Corresponding to 2-Row Diagrams in the Space of Square-Free Symmetric Forms

2007

Dokl. Math.

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.

Section: Proposition 1 the Representation Of The Infinite Symmetric mentioning

confidence: 94%

Induced representations of the infinite symmetric group and their spectral theory

2007

Dokl. Math.

“…. For clarity, we describe the corresponding invariant subspaces (see [9], and also [12], for details in the symmetric case for any k). In this case, are the invariant subspaces corresponding to the irreducible representations π (n) , π (n−1,1) , and π (n−2,2) , respectively.…”

Section: Examplesmentioning

confidence: 99%

On the Decomposition of Tensor Representations of Symmetric Groups

Nikitin

2018

Algebr Represent Theor

Self Cite

“…In this section, we describe the so-called tensor model of irreducible representations of the symmetric groups corresponding to two-row diagrams, which was suggested by the second author and investigated in [3]. In particular, using this model, one can construct the so-called concomitant representation, an irreducible representation of S ∞ associated in a natural way with its factor representation.…”

Section: The Tensor Model Of Two-row Representations and The Concomitmentioning

confidence: 99%

“…Theorem 2. [3] (1) Let k n/2. The representation of the symmetric group S n in the space A 0 n,k (and in the space A 0 n,n−k ) is equivalent to the irreducible representation π n−k,k corresponding to the two-row diagram λ n,k = (n − k, k) with rows of lengths n − k and k.…”

Section: Finite Casementioning

confidence: 99%

“…In this paper, we also consider the socalled "tensor" model of two-row representations of the symmetric groups, which was suggested by the second author and investigated in [3]. In a sense, it plays the role of a "bridge" between the tableau and dynamical models of factor representations of S ∞ and yields an interpretation of another two representations arising in these models-the "diagonal" representation in the dynamical model and the "concomitant" representation in the tableau model.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On different models of representations of the infinite symmetric group

Advances in Applied Mathematics

2006

We present an explicit description of the isomorphism between two models of finite factor representations of the infinite symmetric group: the tableau model in the space of functions on Young bitableaux and the dynamical model in the space of functions on pairs of Bernoulli sequences. The main tool used is the Fourier transform on the symmetric groups. We also start the investigation of the so-called tensor model of two-row representations of the symmetric groups, which plays an intermediate role between the tableau and dynamical models, and show its relations to both these models.