We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem.
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References
A. V. Gnedin, “The representation of composition structures,” Ann. Probab., 25, No. 3, 1437–1450 (1997).
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).
M. V. Karev and P. P. Nikitin, “The boundary of the refined Kingman graph,” Zap. Nauchn. Semin. POMI, 468, 58–74 (2018).
S. V. Kerov, “Combinatorial examples in the theory of AF-algebras,” Zap. Nauchn. Semin. LOMI, 172, 55–67 (1989).
S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, Transl. Math. Monographs, 219, Amer. Math. Soc., Providence, Rhode Island, 2003.
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. V. Kerov and A. M. Vershik, “Locally semisimple algebras. Combinatorial theory and the K0-functor,” J. Sov. Math., 38, No. 2, 1701–1733 (1987).
S. Kerov and A. Vershik, “The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory K0-functor of AF-algebras,” in: A. M. Vershik and D. P. Zhelobenko (eds.), Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., 7 (1990), pp. 39–117.
J. F. C. Kingman, “The representation of partition structures,” J. London Math. Soc. (2), 18, 374–380 (1978).
K. Luoto, S. Mykytiuk, and S. van Willigenburg, An Introduction to Quasisymmetric Schur Functions, Springer, New York (2013).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 498, 2020, pp. 38–54.
Translated by N. V. Tsilevich.
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Safonkin, N.A. Semifinite Harmonic Functions on the Gnedin–Kingman Graph. J Math Sci 255, 132–142 (2021). https://doi.org/10.1007/s10958-021-05356-9
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DOI: https://doi.org/10.1007/s10958-021-05356-9