We study semifinite harmonic functions on arbitrary branching graphs. We give a detailed exposition of an algebraic method which allows one to classify semifinite indecomposable harmonic functions on some multiplicative branching graphs. It was suggested by A. Wassermann in terms of operator algebras, but we rephrase, clarify, and simplify the main arguments working only with combinatorial objects. This work was inspired by the theory of traceable factor representations of the infinite symmetric group S(∞).
B. Blackadar, Operator Algebras, Springer-Verlag, Berlin (2006).
A. Borodin and G. Olshanski, Representations of the Infinite Symmetric Group, Cambridge Univ. Press, Cambridge (2017).
R. P. Boyer, “Characters of the infinite symplectic group — a Riesz ring approach,” J. Funct. Anal., 70, No. 2, 357–387 (1987).
R. P. Boyer, “Infinite traces of AF-algebras and characters of Up8q,” J. Operator Theory, 9, No. 2, 205–236 (1983).
O. Bratteli, “Inductive limits of finite dimensional C˚-algebras,” Trans. Amer. Math. Soc., 171, 195–234 (1972).
A. Gnedin and G. Olshanski, “Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams,” Int. Math. Res. Not., 2006, Art. ID 51968 (2006).
S. Kerov and A. Vershik, “The Grothendieck group of the infinite symmetric group and symmetric functions (with the elements of the theory of K0-functor of AF-algebras),” in: Representation of Lie Groups and Related Topics, Gordon and Breach, New York (1990), pp. 36–114.
S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, Amer. Math. Soc., Providence, Rhode Island (2003).
S. V. Kerov and A. M. Vershik, “Asymptotic theory of the characters of a symmetric group,” Funkts. Anal. Prilozhen., 15, No. 4, 15–27 (1981).
S. V. Kerov and A. M. Vershik, “Locally semisimple algebras. Combinatorial theory and the K0-functor,” Itogi Nauki i Tekhniki, Akad. Nauk SSSR, VINITI, Moscow (1985), pp. 3–56.
S. V. Kerov and A. M. Vershik, “The K-functor (Grothendieck group) of the infinite symmetric group,” Zap. Nauchn. Semin. LOMI, 123, 126–151 (1983).
S. Kerov, A. Okounkov, and G. Olshanski, “The boundary of the Young graph with Jack edge multiplicities,” Int. Math. Res. Not., 1998, No. 4, 173–199 (1998).
K. Matveev, “Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture,” arXiv:1711.06939[math.RT].
N. A. Safonkin, “Semifinite harmonic functions on the Gnedin–Kingman graph,” J. Math. Sci., 255, 132–142 (2021).
R. P. Stanley, Enumerative Combinatorics, Vol. 1, 2nd edition, Cambridge Univ. Press, Cambridge (2012).
S, . Strătilă and D. Voiculescu, Representations of AF-Algebras and of the Group Up8q, Springer-Verlag, Berlin–New York (1975).
A. M. Vershik and P. P. Nikitin, “Description of the characters and factor representations of the infinite symmetric inverse semigroup,” Funct. Anal. Appl., 45, No. 1, 13–24 (2011).
A. J. Wassermann, “Automorphic actions of compact groups on operator algebras,” Ph.D. thesis, University of Pennsylvania (1981); https://repository.upenn.edu/dissertations/AAI8127086/.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 507, 2021, pp. 114–139.
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Safonkin, N.A. Semifinite Harmonic Functions on Branching Graphs. J Math Sci 261, 669–686 (2022). https://doi.org/10.1007/s10958-022-05779-y