( q, t)-Deformations of multivariate hook product formulae
Soichi Okada
DOI: 10.1007/s10801-010-0221-0
Abstract
We generalize multivariate hook product formulae for P-partitions. We use Macdonald symmetric functions to prove a ( q, t)-deformation of Gansner's hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a d-complete poset, we present a conjectural ( q, t)-deformation of Peterson-Proctor's hook product formula.
Pages: 399–416
Keywords: keywords hook product formula; reverse plane partition; Macdonald symmetric functions; $P$-partition; $d$-complete poset
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References
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2. Foda, O., Wheeler, M.: Hall-Littlewood plane partitions and KP. Int. Math. Res. Not. 2009, 2597- 2619 (2009)
3. Foda, O., Wheeler, M., Zuparik, M.: On free fermions and plane partitions. J. Algebra 321, 3249-3273 (2009)
4. Gansner, E.R.: The Hillman-Grassl correspondence and the enumeration of reverse plane partitions. J. Comb. Theory Ser. A 30, 71-89 (1981)
5. Ishikawa, M.: Private communication dated Sep. 17, 2009
6. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Univ. Press, Oxford (1995)
7. Nakada, K.: q-Hook formula for a generalized Young diagram. Preprint
8. Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581-603 (2003)
9. Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and classical crystals. In: The Unity of Mathematics. Progr. Math., vol. 244, pp. 597-618. Birkhäuser, Boston (2006)
10. Proctor, R.A.: Dynkin diagram classification of λ-minuscule Bruhat lattices and d-complete posets. J. Algebr. Comb. 9, 61-94 (1999)
11. Proctor, R.A.: Minuscule elements of Weyl groups, the number game, and d-complete posets. J. Al- gebra 213, 272-303 (1999)
12. Proctor, R.A.: Informal description of the hook length property for posets,
13. Sagan, B.E.: Combinatorial proofs of hook generating functions for skew plane partitions. Theor.
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