Partition algebras

Authors:
Tom Halverson;Arun Ram
Affiliations:
Mathematics and Computer Science, Macalester College, Saint Paul, MN;Department of Mathematics, University of Wisconsin-Madison, Madison, WI
Venue:
European Journal of Combinatorics - Special issue on combinatorics and representation theory
Year:
2005

Citing 2
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Abstract

The partition algebra CAk(n) is the centralizer algebra of Sn acting on the k-fold tensor product V×k of its n-dimensional permutation representation V. The partition algebra CAk + ½ (n) is the centralizer algebra of the restriction of V⊗k to Sn-1 ⊆ Sn. We apply the theory of the basic construction (generalized matrix algebras) to the tower of partition algebras CA0(n) ⊆ CA ½ (n) CA1 (n) ⊆ CA1½ (n) ⊆ .... Our main results are: (a) a presentation on generators and relations for CAk(n); (b) a derivation of "Specht modules" from the basic construction; (c) a proof that CAk(n) is semisimple if and only if k ≤ (n + 1)/2 (except for a few special cases); (d) Murphy elements for CAk(n); and (e) an exposition on the theory of the basic construction and semisimple algebras.