TY - JOUR

T1 - Induced representations of the infinite symmetric group

AU - Tsilevich, Natalia V.

AU - Vershik, A. M.

PY - 2007/10

Y1 - 2007/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=42949179137&partnerID=8YFLogxK

U2 - 10.4310/PAMQ.2007.v3.n4.a7

DO - 10.4310/PAMQ.2007.v3.n4.a7

M3 - Article

AN - SCOPUS:42949179137

VL - 3

SP - 1005

EP - 1026

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

SN - 1558-8599

IS - 4

ER -