Abstract

Let X denote a finite nonempty set, and let W denote a matrix whose rows and columns are indexed by X and whose entries belong to some field K. We study three planar algebras related to W. Briefly, a planar algebra is a graded vector space V = ∪_{n in Z⁺∪{+,−}}V_n which is closed under “planar” operators.

The first planar algebra which we study, F^W = ∪F_n^W, is defined by the group theoretic properties of W. For n in Z⁺, F_n^W is the vector space of functions from Xⁿ to K which are constant on the Aut(W)-orbits of Xⁿ, and F₊^W, F₋^W are identified with K. The second planar algebra, P^W = ∪P_n^W, is the planar algebra generated W. We define it combinatorially: P_n^W is spanned by functions from Xⁿ to K defined via statistical mechanical sums on certain planar open graphs. The third planar algebra, O^W = ∪O_n^W, differs from P^W only in that the open graphs defining the functions need not be planar.

It turns out that P^W ⊆O^W ⊆F^W. We show that P^W = O^W if and only if P₄^W contains a single special function known as the “transposition”. We show that O^W = F^W whenever |X|! is not divisible by the characteristic of K.

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