On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State

Abstract

We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at z i = 1 are positive in n. At n = 1, they coincide with the the Razumov-Stroganov integers counting alternating sign matrices.We derive the CFT modular invariant partition functions labelled by CoxeterDynkin diagrams using the representation theory of the affine Hecke algebras.

Polynomial Representations of the Hecke Algebra of the Symmetric Group

“…The normalization factors have been chosen in such a way that the non-zero specializations are all equal. This proves (12). The space Pol d e t λ H n coincides with the space ∆ q λ H n .…”

“…The space Pol d e t λ H n coincides with the space ∆ q λ H n . Since both e t λ and ∆ −1 to obtain a constant scalar product in (12). In fact, Young defines a "tableau function" f (t) in [32, p.458] , such that the normalization factor specializes, for q 1 = 1, q 2 = −1, to the quotient f (t)/f (t ′ ) for a pair of tableaux differing by a simple transposition (see also [10, p.312], and [25, p.47] for a similar function).…”

“…which are known to physicists [12,5]. In terms of the Young basis, by computing five specializations , one obtains that the first one is equal to …”

Polynomial solutions ofqKZ equation and ground state of XXZ spin chain at Δ = −1/2

“…2−ε 1 ,4−ε 2 ,...,2n−2−ε n−1 ,2n−1 = π: #{even openings of π}=r−1 ξ π (38) as conjectured in the paper [26]. In the special case r = 1 or r = n, the sum in the last right hand side of Eq.…”

Loop model with mixed boundary conditions,qKZ equation and alternating sign matrices

“…In [19] it was observed that in the case of periodic boundary conditions, the components of solutions of the qKZ equation in which the variables z i are set to 1 are polynomials with positive coefficients in the variable: (1) τ = −q − q −1 in the case s = q 6 ; and (2) τ ′ = q 1/2 + q −1/2 in the case s = q 3 . See also [25] for an interesting conjecture on the combinatorial interpretation of the sum of homogeneous components for arbitrary q in the case s = q 6 .…”