Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations

Details Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations

2007-12-21

arxiv.org/abs/0712.3584v2

J. Combin. Theory A 116 (2009), 772--794

Mathematics

Combinatorics

27 pages, 29 eps figures, section rewritten and reference added

Scientific paper

10.1016/j.jcta.2008.11.008

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik--Zamolodchikov equation with reflecting boundaries in the Dyck path representation. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of $\tau^2$-weighted punctured cyclically symmetric transpose complement plane partitions where $\tau=-(q+q^{-1})$. In the cases of no or minimal punctures, we prove that these generating functions coincide with $\tau^2$-enumerations of vertically symmetric alternating sign matrices and modifications thereof.

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