Geodesic
On realizations of representations of the infinite symmetric group
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 110-117 
N. I. Nessonov. On realizations of representations of the infinite symmetric group. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 110-117. http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a7/
@article{ZNSL_2012_403_a7,
     author = {N. I. Nessonov},
     title = {On realizations of representations of the infinite symmetric group},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {110--117},
     year = {2012},
     volume = {403},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a7/}
}
TY  - JOUR
AU  - N. I. Nessonov
TI  - On realizations of representations of the infinite symmetric group
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2012
SP  - 110
EP  - 117
VL  - 403
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a7/
LA  - ru
ID  - ZNSL_2012_403_a7
ER  - 
%0 Journal Article
%A N. I. Nessonov
%T On realizations of representations of the infinite symmetric group
%J Zapiski Nauchnykh Seminarov POMI
%D 2012
%P 110-117
%V 403
%U http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a7/
%G ru
%F ZNSL_2012_403_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Denote by $\mathbb N$ the set of positive integers $\{1,2,\dots\}$. Let $\mathfrak S_\mathbb X$ stand for the group of all finite permutations of the set $\mathbb X=-\mathbb N\cup\mathbb N$. Consider the subgroups $$ \mathfrak S_\mathbb N=\{s\in\mathfrak S_\mathbb X\colon s(-k)=-k\text{ for all }k\in\mathbb N\} $$ and $$\mathfrak D=\{s\in\mathfrak S_\mathbb X\colon -s(k)=s(-k)\text{ and }s(\mathbb N)=\mathbb N\}. $$ Given a spherical representation $\pi$ of the pair $(\mathfrak S_\mathbb N\cdot\mathfrak S_{-\mathbb N},\mathfrak D)$, we construct a spherical representation $\Pi$ of the pair $(\mathfrak S_\mathbb X,\mathfrak D)$ such that the restriction of $\Pi$ to the group $\mathfrak S_\mathbb N\cdot\mathfrak S_{-\mathbb N}$ coincides with $\pi$.

[1] A. M. Vershik, S. V. Kerov, “Kharaktery i faktor predstavleniya beskonechnoi simmetricheskoi gruppy”, DAN SSSR, 257:5 (1981), 1037–1040 | MR | Zbl

[2] A. V. Dudko, N. I. Nessonov, “A description of characters on the infinite wreath product”, Methods Funct. Anal. Topology, 13:4 (2007), 301–317 | MR | Zbl

[3] Yu. A. Neretin, A remark on representations of infinite symmetric groups, arXiv: 1204.4198

[4] A. Yu. Okounkov, “The Thoma theorem and representation of the infinite bisymmetric group”, Funct. Anal. Appl., 28:2 (1994), 100–107 | DOI | MR | Zbl

[5] G. I. Olshanskii, “Unitarnye predstavleniya $(G,K)$-par, svyazannykh s beskonechnoi simmetricheskoi gruppoi $S(\infty)$”, Algebra i analiz, 1:4 (1989), 178–209 | MR | Zbl

[6] E. Thoma, “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe”, Math. Z., 85:1 (1964), 40–61 | DOI | MR | Zbl