M. Parvathi

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].

Signed Partition Algebras

Abstract The partition algebra P k (x) has been defined by 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, 1993. River edge, NJ: World Scientific, pp. 259–267], as one having a basis consisting of partitions of 2k objects. We introduce a new class of algebras called “Signed Partition Algebras,” denoted by , which contain partition algebras as subalgebras. The signed partition algebras are a generalization of the partition algebras and signed Brauer algebras [Brauer, R. (1937). On algebras which are connected with the semisimple continuous groups. Ann. Math. 38:854–872; Parvathi, M., Kamaraj, M. (1998). Signed Brauers algebras. Comm. Algebra 26(3):839–855]. We also give a description in terms of generators and relations. Further we show that, when G = Z 2 the algebra , “The Algebra of G-relations,” introduced by Kodiyalam et al. [Kodiyalam, V., Srinivasan, R., Sunder, V. S. (2000). The algebra of G-relations. Proc. Indian Acad. Sci. (Math. Sci.) 110(3):263–292], may be realized as the centralizer of certain subgroup of the symmetric group action on V ⊗k , and that contains the signed partition algebra , as a subalgebra.