A q-Analog of the Hook Walk Algorithm for Random Young Tableaux
S. Kerov
DOI: 10.1023/A:1022423901412
Abstract
A probabilistic algorithm, called the q-hook walk, is defined. For a given Young diagram, it produces a new one by adding a random box with probabilities, depending on a positive parameter q. The corresponding Markov chain in the space of infinite Young tableaux is closely related to the knot invariant of Jones, constructed via traces of Hecke algebras. For q = 1, the algorithm is essentially the hook walk of Greene, Nijenhuis, and Wilf. The q-hook formula and a q-deformation of Young graph are also considered.
Pages: 383–396
Keywords: Young diagram; random Young tableau; hook formula; q-analog; Hecke algebra
Full Text: PDF
References
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2. C. Greene, A. Nijenhuis, and H. Wilf, "Another probabilistic method in the theory of Young tableaux," J. Combin. Theory Series A, 37 (1984), 127-135.
3. J.S. Frame, G. de B. Robinson, and R.M. Thrall, "The hook graphs of the symmetric group" Can. J. Math. 6 (1954), 316-324.
4. I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
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6. S. Kerov, A. Vershik, "Characters and realizations of representations of an infinite-dimensional Hecke algebra, and knot invariants," Sou Math. Dokl. 38 (1989), 134-137.
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2. C. Greene, A. Nijenhuis, and H. Wilf, "Another probabilistic method in the theory of Young tableaux," J. Combin. Theory Series A, 37 (1984), 127-135.
3. J.S. Frame, G. de B. Robinson, and R.M. Thrall, "The hook graphs of the symmetric group" Can. J. Math. 6 (1954), 316-324.
4. I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
5. B. Pittel, "On growing a random Young tableau," J. Combin. Theory Series A 41 (1986), 278-285.
6. S. Kerov, A. Vershik, "Characters and realizations of representations of an infinite-dimensional Hecke algebra, and knot invariants," Sou Math. Dokl. 38 (1989), 134-137.
7. A.M. Vershik, "Hook formula and related identities," Zapiski Nauchnyckh Seminarov LOMI 172 (1989), 3-20 (in Russian).
8. A.N. Kirillov, "Lagrange identity and the hook formula," Zapiski Nauchnyckh Seminarov LOMI, 172 (1989), 78-87 (in Russian).