Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials

TL;DR

This paper establishes higher rank sum rules for the groundstate of the O(1) loop model, connecting quantum algebra solutions to combinatorial objects like alternating sign matrices and Joseph polynomials.

Contribution

It constructs polynomial solutions to quantum Knizhnik--Zamolodchikov equations that generalize Razumov--Stroganov sum rules and links them to combinatorial and geometric structures.

Findings

Weighted sums of groundstate components yield generalized combinatorial numbers.

Constructed solutions relate to quantum Hall wave functions at filling fraction 1/k.

Identified solutions with extended Joseph polynomials in the rational limit.

Abstract

We prove higher rank analogues of the Razumov--Stroganov sum rule for the groundstate of the O(1) loop model on a semi-infinite cylinder: we show that a weighted sum of components of the groundstate of the A_{k-1} IRF model yields integers that generalize the numbers of alternating sign matrices. This is done by constructing minimal polynomial solutions of the level 1 U_q(\hat{sl(k)}) quantum Knizhnik--Zamolodchikov equations, which may also be interpreted as quantum incompressible q-deformations of fractional quantum Hall effect wave functions at filling fraction nu=1/k. In addition to the generalized Razumov--Stroganov point q=-e^{i pi/k+1}, another combinatorially interesting point is reached in the rational limit q -> -1, where we identify the solution with extended Joseph polynomials associated to the geometry of upper triangular matrices with vanishing k-th power.