Geodesic
Characters of the infinite alternating group ${\mathfrak{A}}_{\mathbb{N}}$ and ${\mathbb{N}}$-graded quotient graphs over involution
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXV, Tome 528 (2023), pp. 91-106 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In 1964, German mathematician E. Thoma published the complete list of extreme characters of the infinite symmetric and alternating groups; the translation of this work and the commentary on it have been published in the current volume. Thoma has deduced the classification of extreme characters of the infinite alternating group $\mathfrak{A}_{\mathbb{N}}$ from the corresponding result for the symmetric group and general properties of countable groups that he has shown in another work. We suggest another, more direct proof of this result using different technique, – we consider the graph (Bratelli diagram), which may be viewed as a quotient of the Young graph by its natural involution. Effectively, we prove a general result, namely, given the set of ergodic measures on a graph with an involution, we explain how to describe the set of ergodic central measures on the quotient graph. The problems of how the traces (the characters) change after various changes of a graph, have not been sufficiently explored. 
@article{ZNSL_2023_528_a5,
     author = {A. M. Vershik and V. N. Ivanov},
     title = {Characters of the infinite alternating group ${\mathfrak{A}}_{\mathbb{N}}$ and ${\mathbb{N}}$-graded quotient graphs over involution},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {91--106},
     year = {2023},
     volume = {528},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a5/}
}
TY  - JOUR
AU  - A. M. Vershik
AU  - V. N. Ivanov
TI  - Characters of the infinite alternating group ${\mathfrak{A}}_{\mathbb{N}}$ and ${\mathbb{N}}$-graded quotient graphs over involution
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2023
SP  - 91
EP  - 106
VL  - 528
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a5/
LA  - en
ID  - ZNSL_2023_528_a5
ER  - 
%0 Journal Article
%A A. M. Vershik
%A V. N. Ivanov
%T Characters of the infinite alternating group ${\mathfrak{A}}_{\mathbb{N}}$ and ${\mathbb{N}}$-graded quotient graphs over involution
%J Zapiski Nauchnykh Seminarov POMI
%D 2023
%P 91-106
%V 528
%U http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a5/
%G en
%F ZNSL_2023_528_a5
A. M. Vershik; V. N. Ivanov. Characters of the infinite alternating group ${\mathfrak{A}}_{\mathbb{N}}$ and ${\mathbb{N}}$-graded quotient graphs over involution. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXV, Tome 528 (2023), pp. 91-106. http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a5/

[1] T. Geetha, A. Prasad, “Comparison of Gelfand–Tsetlin bases for alternating and symmetric groups”, Algebras and Representation Theory, 21 (2018), 131–143

[2] G. D. James, A. Kerber, The Representation Theory of the Symmetric Group, Cambridge University Press, Cambridge, 1984

[3] S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, Transl. Math. Monogr., 219, 2003

[4] B. F. Logan, L. A. Shepp, “A variational problem for random Young tableaux”, Advances in Mathematics, 26:2 (1977), 206–222

[5] E. Thoma, “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe”, Mathematische Zeitschrift, 85 (1964), 40–61

[6] E. Thoma, “Über unitäre Darstellungen abzählbarer, diskreter Gruppen”, Mathematische Annalen, 153 (1964), 111–138

[7] S. Thomas, “Characters of inductive limits of finite alternating groups”, Ergodic Theory and Dynamical Systems, 40:4 (2020), 1068–1082

[8] A. M. Vershik, “Spectrum and absolute of the graph of two-row Young diagrams”, Zap. Nauchn. Semin. POMI, 517, 2022, 55–69

[9] A. M. Vershik, S. V. Kerov, “Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux”, Sov. Math., Dokl., 18 (1977), 527–531

[10] A. M. Vershik, S. V. Kerov, “Characters and factor representations of the infinite symmetric group”, Sov. Math., Dokl., 23 (1981), 389–392

[11] A. M. Vershik, S. V. Kerov, “Asymptotic theory of characters of the symmetric group”, Funct. Anal. Appl., 15 (1982), 246–255 \pagebreak

[12] A. M. Vershik, S. V. Kerov, “The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of $K_ 0$-functor of AF-algebras)”, Representation of Lie groups and related topics, Adv. Stud. Contemp. Math., 7, 1990, 39–117