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 Brauer's 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.

Key Words:

• Partition algebra
• Centralizer algebra
• Direct product
• Symmetric group

1991 AMS Subject Classification:

• 16S99

Acknowledgment

The author would like to express her sincere thanks to the Referee for his useful comments and suggestions for the improvement of the paper.

Notes

^#Communicated by A. Facchini