Skip to main content
SpringerLink
Log in

Semifinite Harmonic Functions on the Zigzag Graph

  • Research Articles
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

We study semifinite harmonic functions on the zigzag graph, which corresponds to the Pieri rule for the fundamental quasisymmetric functions \(\{F_{\lambda}\}\). The main problem, which we solve here, is to classify the indecomposable semifinite harmonic functions on this graph. We show that these functions are in a natural bijective correspondence with some combinatorial data, the so-called semifinite zigzag growth models. Furthermore, we describe an explicit construction that produces a semifinite indecomposable harmonic function from every semifinite zigzag growth model. We also establish a semifinite analogue of the Vershik–Kerov ring theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

Notes

  1. The papers [14], [3], and [16] are written in the language of operator algebras, while we work with combinatorial objects and talk about harmonic functions on branching graphs instead of traces on AF-algebras.

  2. By an embedding \(f\colon \Gamma _1\rightarrow \Gamma _2\) of graded graphs we mean an injective map between the vertex sets such that, for any \( \lambda ,\mu\in \Gamma _1\), we have \( \lambda \nearrow \mu\) if and only if \(f( \lambda )\nearrow f(\mu)\).

References

  1. R. P. Boyer, “Infinite traces of AF-algebras and characters of U\((\infty)\)”, J. Operator Theory, 9:2 (1983), 205–236.

    MATH  Google Scholar 

  2. R. P. Boyer, “Characters of the infinite symplectic group—a Riesz ring approach”, J. Funct. Anal., 70:2 (1987), 357–387.

    Article  MATH  Google Scholar 

  3. O. Bratteli, “Inductive limits of finite dimensional \(C^*\)-algebras”, Trans. Amer. Math. Soc., 171 (1972), 195–234.

    MATH  Google Scholar 

  4. I. M. Gessel, “Multipartite \(P\)-partitions and inner products of skew Schur functions”, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., no. 34, Amer. Math. Soc., Providence, RI, 1984, 289–317.

    Chapter  Google Scholar 

  5. 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. ID51968.

    MATH  Google Scholar 

  6. S. V. Kerov, “Combinatorial examples in the theory of AF-algebras”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 172 (1989), 55–67.

    MATH  Google Scholar 

  7. S. V. Kerov and A. M. Vershik, “The \(K\)-functor (Grothendieck group) of the infinite symmetric group”, Zap. Nauchn. Sem. LOMI, 123 (1983), 126–251.

    MATH  Google Scholar 

  8. S. V. Kerov and A. M. Vershik, “Locally semisimple algebras. Combinatorial theory and the \(K_0\)-functor”, Itogi Nauki i Tekhniki. Current problems in mathematics. Newest results, no. 26, 1985, 3–56.

    Google Scholar 

  9. S. Kerov and A. Vershik, “The Grothendieck group of the infinite symmetric group and symmetric functions (with the elements of the \(K_0\)-functor theory of AF-algebras)”, Representation of Lie groups and related topics, Adv. Stud. Contemp. Math, no. 7, Gordon and Breach, New York, 1990, 36–114.

    Google Scholar 

  10. K. Luoto, S. Mykytiuk, and S. van Willigenburg, An Introduction to Quasisymmetric Schur Functions, SpringerBriefs in Mathematics, Springer, New York, 2013.

    Book  MATH  Google Scholar 

  11. N. A. Safonkin, “Semifinite harmonic functions on branching graphs”, Zap. Nauchn. Sem. POMI, 507 (2021), 114–139; arXiv: 2108.07850.

    Google Scholar 

  12. N. Safonkin, Semifinite harmonic functions on the zigzag graph, arXiv: 2110.01508.

  13. R. P. Stanley, Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, no. 62, Cambridge University Press, Cambridge, 2009.

    Google Scholar 

  14. Ş. Strătilă and D. Voiculescu, Representations of AF-Algebras and of the Group \(U(\infty)\) , Lecture Notes in Mathematics, no. 486, Springer-Verlag, Berlin–New York, 1975.

    Book  MATH  Google Scholar 

  15. P. Tarrago, “Zigzag diagrams and Martin boundary”, Ann. Probab., 46:5 (2018), 2562–2620.

    Article  MATH  Google Scholar 

  16. A. J. Wassermann, Automorphic actions of compact groups on operator algebras, University of Pennsylvania, 1981.

    Google Scholar 

Download references

Acknowledgments

I am deeply grateful to Grigori Olshanski for many useful comments and stimulating discussions. I would like to thank Pavel Nikitin for careful reading of the paper and helpful discussions.

Funding

This work was supported in part by the Simons Foundation and by the Basic Research Program at the National Research University Higher School of Economics.

Author information

Authors and Affiliations

  1. Skolkovo Institute of Science and Technology, Moscow, Russia

    N. A. Safonkin

  2. National Research University Higher School of Economics, Moscow, Russia

    N. A. Safonkin

Authors
  1. N. A. Safonkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Safonkin.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 52–74 https://doi.org/10.4213/faa4013.

Translated by N. A. Safonkin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Safonkin, N.A. Semifinite Harmonic Functions on the Zigzag Graph. Funct Anal Its Appl 56, 199–215 (2022). https://doi.org/10.1134/S0016266322030042

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0016266322030042

Keywords

Search

Navigation