Subalgebras of partition algebras

Subalgebras of partition algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

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 1 2 >0 , the following are subalgebras of the partition algebra Ak n : Sk = span d Ak | pn d =k , Pk n = span d Ak | d  is planar , Bk n = span d Ak | all blocks of  d  have size 2 , Tk n = span d Ak | d  is planar and all blocks of  d  have size 2 . The algebra Sk is the group algebra of the symmetric group, Pk n is the planar partition algebra, Bk n is the Bruer algebra, and Tk n is the Temperley-Lieb algebra. Examples of set partitions in these algebras are

P7 ,

P 6+ 1 2 ,

B7 ,

T7 ,

S7 .

If B is a block of a set partition d define κ B = #  of top vertices in  B - #  of bottom vertices in  B and let A k,r,p = l=0 r/p -1 d Ak | for all blocks  B  of  d,κ B =l r/p modr Then A k,r,p n =span xd | d A k,r,p is a subalgebra of Ak n . Then A k,r,p n A k,r,1 n , A k,1,1 n = Ak n , and A k,,1 =span d Ak | κ B =0  for all blocks  B  of  d does not depend on the parameter n.

Let fr = p 1 2 p 3 2 p r-1 2 p1 p2 pr p 1 2 p 3 2 p r-1 2 =

r 1

The algebra A k,r,1 n is generated by s1 ,, s k-1 , p 3 2 and fr .

Is CC Ak rpn = Ak rpn ×/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)

page history