Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices

P. Di Francesco (1) , Paul Zinn-Justin (2)

1 SPhT - Service de Physique Théorique
2 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

We present multiresidue formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco '06] connecting the above to generating polynomials for weighted Totally Symmetric Self-Complementary Plane Partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Combinatoire [math.CO]

P. Di Francesco, Paul Zinn-Justin. Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices. Theoretical and Mathematical Physics, 2008, 154, pp.331-348. ⟨hal-00285943⟩