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.
Article PDF
Similar content being viewed by others
References
C. Greene, A. Nijenhuis, and H. Wilf, “A probabilistic proof of a formula for the number of Young tableaux of a given shape,” Adv. Math., 31 (1979), 104–109.
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.
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.
I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
B. Pittel, “On growing a random Young tableau,” J. Combin. Theory Series A 41 (1986), 278–285.
S. Kerov, A. Vershik, “Characters and realizations of representations of an infinite-dimensional Hecke algebra, and knot invariants,” Sov. Math. Dokl. 38 (1989), 134–137.
A.M. Vershik, “Hook formula and related identities,” Zapiski Nauchnyckh Seminarov LOMI 172 (1989), 3–20 (in Russian).
A.N. Kirillov, “Lagrange identity and the hook formula,” Zapiski Nauchnyckh Seminarov LOMI, 172 (1989), 78–87 (in Russian).
Author information
Authors and Affiliations
Additional information
S. Kerov: Supported by a grant from CRM (Université de Montréal), during its Operator Algebras year
Rights and permissions
About this article
Cite this article
Kerov, S. A q-Analog of the Hook Walk Algorithm for Random Young Tableaux. Journal of Algebraic Combinatorics 2, 383–396 (1993). https://doi.org/10.1023/A:1022423901412
Issue Date:
DOI: https://doi.org/10.1023/A:1022423901412