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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∈ℤ+∪{+,−}𝒱n which is
closed under “planar” operators.
The first planar algebra which we study, ℱW = ∪ℱnW, is defined by the
group theoretic properties of W. For n ∈ ℤ+, ℱnW 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, 𝒫W = ∪𝒫nW,
is the planar algebra generated W. We define it combinatorially: 𝒫nW is
spanned by functions from Xn to 𝕂 defined via statistical mechanical sums on
certain planar open graphs. The third planar algebra, 𝒪W = ∪𝒪nW, differs
from 𝒫W only in that the open graphs defining the functions need not be
planar.
It turns out that 𝒫W ⊆𝒪W ⊆ℱW. We show that 𝒫W = 𝒪W if and only if
𝒫4W contains a single special function known as the “transposition”. We
show that 𝒪W = ℱW whenever |X|! is not divisible by the characteristic of
𝕂.
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