Mathematical Sciences Research Institute - The Razumov-Stroganov conjecture: loop gas, alternating sign matrices, plane partitions and orbital varieties

We review recent progress towards a proof of the Razumov-Stroganov (RS)
conjecture that relates correlations of a statistical model of loops on a
semi-infinite cylinder to alternating sign matrices (ASM). The latter are known
since the work of Kuperberg to be in bijection with the configurations of
the square ice or six-vertex model.

We first prove a sum rule for the correlations of an integrable inhomogeneous
version of the loop model, expressing it in terms of the partition function
of the inhomogeneous ice model. We next prove another sum rule for a
q-deformation of the loop model, expressing it in terms of generating functions
for weighted totally symmetric self-complementary plane partitions (TSSCPP).
Both proofs rely on a reformulation of the correlators of the loop model in
terms of polynomial solutions of the quantum Knizhnik-Zamolodchikov
(qKZ) equation.

This gives a new connection between ASM and TSSCPP. We also prove a
different version of the RS conjecture that interprets solutions to qKZ at
$q=-1$ in terms of the degree of the irreducible components of the variety
of complex triangular matrices with vanishing square.

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