G-Vertex Colored Partition Algebras as Centralizer Algebras of Direct Products#

Abstract

The partition algebras P k (x) have been defined in Martin [Martin, P. (1990). Representations of graph temperley Lieb algebras. Pupl. Res. Inst. Math. Sci. 26:485–503] and Jones [Jones, V. F. R. (1993). The Potts model and the symmetric group. In: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebra, Kyuzeso. River edge, NJ: World Scientific, pp. 259–267, 1994]. We introduce a new class of algebras for every group G called “G-Vertex Colored Partition Algebras,” denoted by P k (x, G), which contain partition algebras P k (x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra P k (n, G) is the centralizer algebra of an action of the direct product S n × G on tensor powers of its permutation module. Further we show that these algebras P k (x, G) contain as subalgebras the “G-Colored Partition Algebras P k (x;G)” introduced in Bloss [Bloss, M. (2003). G-colored partition algebras as centralizer algebras of wreath products. J. Algebra 265:690–710].