Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu
Last updates: 20 June 2010
Subalgebras of partition algebras
A set partition is planar [Jo] if it can be represented as a grpah without edge crossings inside of the rectangle formed by its vertices. For each k∈12ℤ>0, the following are subalgebras of the partition algebra ℂAkn:ℂSk=spand∈Ak|pnd=k,ℂPkn=spand∈Ak|d is planar,ℂBkn=spand∈Ak|all blocks of d have size 2,ℂTkn=spand∈Ak|d is planar and all blocks of d have size 2. The algebra ℂSk is the group algebra of the symmetric group, ℂPkn is the planar partition algebra, ℂBkn is the Bruer algebra, and ℂTkn is the Temperley-Lieb algebra. Examples of set partitions in these algebras are
If B is a block of a set partition d define κB=# of top vertices in B-# of bottom vertices in B and let Ak,r,p=⨆l=0r/p-1d∈Ak|for all blocks B of d,κB=lr/pmodr Then ℂAk,r,pn=spanxd|d∈Ak,r,p is a subalgebra of ℂAkn. Then ℂAk,r,pn⊇ℂAk,r,1n,ℂAk,1,1n=ℂAkn, and ℂAk,∞,1=spand∈Ak|κB=0 for all blocks B of d does not depend on the parameter n.
Let fr=p12p32…pr-12p1p2…prp12p32…pr-12=
The algebra Ak,r,1n is generated by s1,…,sk-1,p32 and fr.