Traces on infinite-dimensional brauer algebras

Abstract

We describe the central measures for the random walk on graded graphs. Using this description, we obtain the list of all finite traces on three infinite-dimensional algebras: on the Brauer algebra, on the partition algebra, and on the walled Brauer algebra. For the first two algebras, these lists coincide with the list of all finite traces of the infinite symmetric group. For the walled Brauer algebra, the list of finite traces coincide with the list of finite traces of the square of the latter group. Brauer algebras and the pascalized graphsConsider the diagonal action of complex orthogonal group O k (C) acts on tensor power V ⊗n of theR. Brauer (see [3,20]) defined an family of finite dimensional algebra Br n (k) the Brauer algebra,depending on complex parameter k and positive integer n. For integer k ≥ k the algebra Br n (k) is isomorphic to centralizer of above described action of O k (C). For k ∈ C | Z ∪ {n . . . } and n fixed these algebras are semisimple and pairwise isomorphic [19]. For now on we consider only these number of parameter k and denote the corresponding algebras by Br n omitting k in the notation. For other k ∈ Z − ∪ {1, . . . n − 1}, k < n the algebra Br n (k) is not semisimple. We shall also study the walled Brauer algebra Br n,m (k), n, m ∈ Z + . The history of this algebras is as follows. V. Turaev ([15]), was first to define it by presentation, he also pointed to the first author that its dimension is (n + m)! and that this algebra resembles the symmetric group. The walled Brauer algebra was defined independently in the work of K. Koike [9], and later it was studied in [1] as the centralizer of the diagonal action of the group GL k (C) on tensor space V ⊗n V * ⊗m . It was clear from its diagrammatic definition that this algebra is subalgebra of the Brauer algebra. The walled Brauer algebra is also semisimple and pairwise isomorphic for the generic k, k ∈ {x ∈ C | x ∈ Z} ∪ {x ∈ Z | |x| ≥ m + n} (see [14] for details). Here we again also consider only these generic values and omit k in the notation, Br n,m = Br n,m (k).P. Martin introduced the partition algebras P art n (k), n ∈ Z + , k ∈ C (see [12]). Algebras P art 2n (k), P art 2n+1 (k) for sufficiently large k ∈ N are isomorphic to the centralizers of the diagonal action of the subgroups S k ⊂ GL k (C), S k−1 ⊂ GL k (C) on tensor space V ⊗n . For the generic k ∈ {x ∈ C | x ∈ Z} ∪ {x ∈ N | x ≥ 2n − 1} and fixed n partition algebras are semisimple and pairwise isomorphic, we will denote them by P art n .Each finite-dimensional algebra concerned include the ideal J (with an appropriate subscript) spanned by all noninvertible standard generators of the corresponding algebra (see [19,14,12]) *