The Martin Boundary of the Young-Fibonacci Lattice
Frederick M. Goodman
and Sergei V. Kerov
DOI: 10.1023/A:1008739619211
Abstract
In this paper we find the Martin boundary for the Young-Fibonacci lattice YF. Along with the lattice of Young diagrams, this is the most interesting example of a differential partially ordered set. The Martin boundary construction provides an explicit Poisson-type integral representation of non-negative harmonic functions on YF. The latter are in a canonical correspondence with a set of traces on the locally semisimple Okada algebra. The set is known to contain all the indecomposable traces. Presumably, all of the traces in the set are indecomposable, though we have no proof of this conjecture. Using an explicit product formula for Okada characters, we derive precise regularity conditions under which a sequence of characters of finite-dimensional Okada algebras converges.
Pages: 17–48
Keywords: differential poset; harmonic function; martin boundary; okada algebra; non-commutative symmetric function
Full Text: PDF
References
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3. S.V. Fomin, “Generalized Robinson-Schensted-Knuth correspondence,” J. Soviet Math. 41 (1988) 979-991.
4. S.V. Fomin, “Duality of graded graphs,” J. Soviet Math 41 (1988) 979-991.
5. S.V. Fomin, “Schensted algorithms for dual graded graphs,” J. Alg. Combin. 4 (1995) 5-45.
6. S. Kerov, A. Okounkov, and G. Olshanski, “The Boundary of Young Graph with Jack Edge Multiplicities,” q-alg/9703037, Kyoto University preprint RIMS-1174.
7. S. Kerov and A. Vershik, “The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of the K0-functor theory of AF-algebras,” Adv. Stud. Contemp. Math., Vol. 7: Representation of Lie groups and related topics, A.M. Vershik and D.P. Zhelobenko (Eds.), Gordon and Breach, 1990, pp. 36-114.
8. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, 1995.
9. S. Okada, “Algebras associated to the Young-Fibonacci lattice,” Trans. Amer. Math. Soc. 346 (1994) 549-568.
10. A.Yu. Okounkov, “Thoma's theorem and representations of infinite bisymmetric group,” Funct. Analysis and its Appl. 28 (1994) 101-107.
11. R.P. Stanley, “Differential Posets,” J. Amer. Math. Soc. 1 (1988) 919-961.
12. R.P. Stanley, Variations on Differential Posets, Invariant Theory and Tableaux, IMA Vol: Math. Appl., D. Stanton (Eds.), Springer Verlag, New York, 1988, pp 145-165.
13. R.P. Stanley, “Further combinatorial properties of two Fibonacci lattices,” European J. Combin. 11 (1990) 181-188.
14. E. Thoma, “Die unzerlegbaren, Positiv-definiten Klassen-funktionen der abzahlbar unendlichen, symmetrischen Gruppe,” Math. Z. 85 (1964) 40-61.
15. A.M. Vershik and S.V. Kerov, “Asymptotic character theory of the symmetric group,” Funct. Analysis and its Appl. 15 (1981) 246-255.