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

We prove higher rank analogues of the Razumov-Stroganov sum rule for the ground state of the O(1) loop model on a semi-infinite cylinder: we show that a weighted sum of components of the ground state 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(sl(k)) quantum Knizhnik-Zamolodchikov equations, which may also be interpreted as quantum incompressible q-deformations of quantum Hall effect wavefunctions at filling fraction v = 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 with the geometry of upper triangular matrices with vanishing kth power.