Some Planar Algebras Related to Graphs
نویسندگان
چکیده
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 = ∪FW n , is defined by the group theoretic properties of W . For n ∈ Z, FW n is the vector space of functions from X to K which are constant on the Aut(W )-orbits of X, and FW + , FW − are identified with K. The second planar algebra, PW = ∪PW n , is the planar algebra generated W . We define it combinatorially: PW n is spanned by functions from X to K defined via statistical mechanical sums on certain planar open graphs. The third planar algebra, OW = ∪OW n , 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 PW 4 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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