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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2005

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

Résumé

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.

Dates et versions

hal-00014289 , version 1 (23-11-2005)

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Citer

P. Di Francesco, Paul Zinn-Justin. Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials. Journal of Physics A: Mathematical and Theoretical, 2005, 38, pp.L815-L822. ⟨hal-00014289⟩
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