Geodesic
The comments to the article by Elmar Thoma: “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abz ählbar unendlichen, symmetrischen Gruppe”
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXV, Tome 528 (2023), pp. 37-46 
A. M. Vershik. The comments to the article by Elmar Thoma: “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abz ählbar unendlichen, symmetrischen Gruppe”. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXV, Tome 528 (2023), pp. 37-46. http://geodesic.mathdoc.fr/item/ZNSL_2023_528_a1/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We comment here the classical article by Elmar Thoma abour caracters of the infinite symmetric group; consider the proof of the main result , which compare with the following papers of different authors who suggested various approves to the theory of the characters and representations of that group as well as related groups.

[1] A. Vershik, A. Shmidt, “Predelnye mery, voznikayuschie v asimptoticheskoi teorii simmetricheskikh grupp. I”, Teor. veroyatn. i ee prim., 22:1 (1977), 72–88 ; 23:1 (1978), 42–54

[2] A. Vershik, “Ravnomernaya algebraicheskaya approksimatsiya operatorov sdviga i umnozheniya”, DAN SSSR, 259:3 (1981), 526–529

[3] A. Vershik, A. Okunkov, “Induktivnyi sposob izlozheniya teorii predstavlenii simmetricheskikh grupp”, UMN, 51:2 (308) (1996), 153–154

[4] M. H. Aissen, A. Edrei, I. J. Schoenberg, A. Whitney, “On the generating functions of nonally positive sequences”, Proc. National Acad. Sci. USA, 37 (1951), 303–307

[5] A. Edrei, “On the generation function of a doubly infinite, totally positive sequence”, Trans. Amer. Math. Soc., 74 (1953), 367–383

[6] A. Edrei, “On the generating functions of totally positive sequences. II”, J. d'Analyse Mathématique, 2 (1952), 104–109

[7] A. Borodin, G. Olshanski, A. Okounkov, “Asymptotics of Plancherel measures for symmetric groups”, J. Amer. Math. Soc., 13:3 (2000), 481–515

[8] A. Yu. Okunkov, “Teorema Toma i predstavleniya beskonechnoi bisimmetricheskoi gruppy”, Funkts. analiz i ego pril., 28:2 (1994), 31–40

[9] A. Vershik, S. Kerov, “Asimptoticheskaya teoriya kharakterov simmetricheskoi gruppy”, Funkts. analiz i ego pril., 15:4 (1981), 15–27

[10] A. Vershik, S. Kerov, “Asimtotika mery Plansherelya simmetricheskoi gruppy i predelnaya forma tablits Yunga”, DAN SSSR, 233:6 (1977), 1024–1027

[11] L. Shepp, B. Logan, “A variational problem for random Young tableaux”, Adv. Math., 26:2 (1977), 206–222

[12] A. Vershik, “Totally nonfree actions and the infinite symmetric group”, Mosc. Math. J., 12:1 (2012), 193–212

[13] S. Kerov, Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, Translations of Mathematical Monographs, 219, 2003

[14] A. Vershik, “Odnomernye tsentralnye mery na numeratsiyakh uporyadochennykh mnozhestv”, Funkts. analiz i ego pril., 56:4 (2022), 17–24

[15] A. Vershik, C. Kerov, “Characters, factor representations and K-functor of the infinite symmetric group”, Operator algebras and group representation (Neptun, 1980), v. II, Monogr. Stud. Math., 18, Pitman, Boston, 1984, 23–32

[16] A. Vershik, S. Kerov, “Lokalno poluprostye algebry. Kombinatornaya teoriya i K0-funktor”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 26, VINITI, M., 1985, 3–56

[17] A. Vershik, “Opisanie invariantnykh mer dlya deistviya nekotorykh beskonechnomernykh grupp”, DAN SSSR, 218:4 (1974), 749–752

[18] S. Stratila, D. Voiculesku, Representation of AF-algebras and of the Group $U(\infty)$, Springer LNM, 486, 1975

[19] A. Vershik, “Tri teoremy o edinstvennosti mery Plansherelya c raznykh pozitsii”, Algebraicheskaya topologiya, kombinatorika i matematicheskaya fizika, Tr. MIAN, 305, MIAN, M., 2019, 71–85