Simplex and Polygon Equations

Abstract

Abstract. It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order". We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of "polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N -simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation.

Hamiltonian integrability of two-component short pulse equations

“…The hierarchy of evolutionary systems (22) possessing ZCR (3) with the matrix X given by (20) is completely determined by the relation (31), where q and r must satisfy the conditions ∂f /∂λ = ∂g/∂λ = 0. If we would have K = 0 in (31), we could immediately write down the recursion operator for the represented hierarchy as R = M L −1 [12].…”

Higher Bruhat and Tamari Orders and their Realizations

“…Let us "localize" them by allowing the maps to depend on additional variables, and such that each of these equations determines a map by taking the variables on the left hand side to those of the right hand side. Then, guided by B(N + 2, N), one can deduce that these maps have to satisfy the (N + 1)-simplex equation [1]. Hence, the latter arises as the consistency condition of a system of localized N-simplex equations.…”

“…Neighboring polygon equations are related by the same kind of "integrability" that connects neighboring simplex equations, and this is what makes them special and promising. The important role of the pentagon equation (some keywords are "multiplicative unitary", "Drinfeld associator", "quantum dilogarithm", see [1] for references) enhances the expectations for its higher analogs. The aforementioned decomposition of Bruhat orders leads to relations between (solutions of) simplex and polygon equations [1].…”

“…We define the N-gon equation as a realization of it, now with maps K  [1]. For N = 5, this is the ubiquitous pentagon equation.…”

“…In this letter, we sketch some of our recent results [1] at the interface between integrable systems and combinatorics. This includes a revision of the relation between higher Bruhat orders and simplex equations [2,3], a decomposition of higher Bruhat orders, the resulting Tamari orders (expected to be equivalent to higher Stasheff -Tamari orders), and a new family of equations associated with the latter.…”