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Article

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

Journal of Combinatorial Theory, Series A. 2009. Vol. 116. P. 772-794.
Pyatov P. N., de Gier J., Zinn-Justin P.

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 of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2-weighted punctured cyclically symmetric transpose complement plane partitions where τ =−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ 2-enumerations of vertically symmetric alternating sign matrices and modifications thereof.