Abstract:
We prove a classification theorem for admissible representation of the Gelfand pair
$$
S(\infty)\times S(\infty)\supset\operatorname{diag}S(\infty)
$$
and two other Gelfand pairs of hyperoctohedral type. We prove that the list of admissible representations given by G. Olshanski is complete. This generalizes Thoma's description of the characters of $S(\infty)$. An explicit construction for representations from a dense subset of the admissible dual was given by G. Olshanski. We construct the remaining representations using an operation we call the mixture of representations.
Citation:
A. Yu. Okounkov, “On representations of the infinite symmetric group”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Zap. Nauchn. Sem. POMI, 240, POMI, St. Petersburg, 1997, 166–228; J. Math. Sci. (New York), 96:5 (1999), 3550–3589
\Bibitem{Oko97}
\by A.~Yu.~Okounkov
\paper On representations of the infinite symmetric group
\inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~II
\serial Zap. Nauchn. Sem. POMI
\yr 1997
\vol 240
\pages 166--228
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl474}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1691646}
\zmath{https://zbmath.org/?q=an:0959.20016}
\transl
\jour J. Math. Sci. (New York)
\yr 1999
\vol 96
\issue 5
\pages 3550--3589
\crossref{https://doi.org/10.1007/BF02175834}
Linking options:
https://www.mathnet.ru/eng/znsl474
https://www.mathnet.ru/eng/znsl/v240/p166
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N. I. Nessonov, “Characters of the Infinite Symmetric Inverse Semigroup”, Funct. Anal. Appl., 54:3 (2020), 179–187
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N. I. Nessonov, “Representations of $\mathfrak{S}_\infty$ admissible with respect to Young subgroups”, Sb. Math., 203:3 (2012), 424–458
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Makoto Yamashita, “On subfactors arising from asymptotic representations of symmetric groups”, Proc. Amer. Math. Soc., 140:1 (2011), 249
A. V. Dudko, N. I. Nessonov, “Characters of projective representations of the infinite generalized symmetric group”, Sb. Math., 199:10 (2008), 1421–1450
Strahov E., “Generalized characters of the symmetric group”, Advances in Mathematics, 212:1 (2007), 109–142
A. V. Dudko, “Tame representations of the group $\mathrm{GL}(\infty,\mathbb F_q)$”, St. Petersburg Math. J., 18:2 (2007), 223–239
Gnedin A., Olshanski G., “Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams”, International Mathematics Research Notices, 2006, 51968
Goodman F.M., Hauschild H., “Affine Birman-Wenzl-Murakami algebras and tangles in the solid torus”, Fund Math, 190 (2006), 77–137
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