Subalgebras of partition algebras

Subalgebras of partition algebras

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 k12>0, the following are subalgebras of the partition algebra Akn: Sk=spandAk|pnd=k,Pkn=spandAk|d  is planar,Bkn=spandAk|all blocks of  d  have size 2,Tkn=spandAk|d  is planar and all blocks of  d  have size 2. The algebra Sk is the group algebra of the symmetric group, Pkn is the planar partition algebra, Bkn is the Bruer algebra, and Tkn is the Temperley-Lieb algebra. Examples of set partitions in these algebras are

P7,

P6+12,

B7,

T7,

S7.

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-1dAk|for all blocks  B  of  d,κB=lr/pmodr Then Ak,r,pn=spanxd|dAk,r,p is a subalgebra of Akn. Then Ak,r,pnAk,r,1n,Ak,1,1n=Akn, and Ak,,1=spandAk|κB=0  for all blocks  B  of  d does not depend on the parameter n.

Let fr=p12p32pr-12p1p2prp12p32pr-12=

r 1

The algebra Ak,r,1n is generated by s1,,sk-1,p32 and fr.

Is CCAkrpn=Akrpn×/p?

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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