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 $k \in \frac{1}{2} ℤ_{> 0},$ the following are subalgebras of the partition algebra $ℂ A_{k} (n) :$ $\begin{array}{rcl} ℂ S_{k} & = & span \{d \in A_{k} | p n (d) = k\}, \\ ℂ P_{k} (n) & = & span \{d \in A_{k} | d is planar\}, \\ ℂ B_{k} (n) & = & span \{d \in A_{k} | all blocks of d have size 2\}, \\ ℂ T_{k} (n) & = & span \{d \in A_{k} | d is planar and all blocks of d have size 2\} . \end{array}$ The algebra $ℂ S_{k}$ is the group algebra of the symmetric group, $ℂ P_{k} (n)$ is the planar partition algebra, $ℂ B_{k} (n)$ is the Bruer algebra, and $ℂ T_{k} (n)$ is the Temperley-Lieb algebra. Examples of set partitions in these algebras are

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 $A_{k, r, p} = ⨆_{l = 0}^{(r / p) - 1} \{d \in A_{k} | for all blocks B of d, κ (B) = l (r / p) (\mod r)\}$ Then $ℂ A_{k, r, p} (n) = span \{x_{d} | d \in A_{k, r, p}\}$ is a subalgebra of $ℂ A_{k} (n) .$ Then $ℂ A_{k, r, p} (n) \supseteq ℂ A_{k, r, 1} (n), ℂ A_{k, 1, 1} (n) = ℂ A_{k} (n),$ and $ℂ A_{k, \infty, 1} = span \{d \in A_{k} | κ (B) = 0 for all blocks B of d\}$ does not depend on the parameter $n .$

Let $f_{r} = p_{\frac{1}{2}} p_{\frac{3}{2}} \dots p_{\frac{(r - 1)}{2}} p_{1} p_{2} \dots p_{r} p_{\frac{1}{2}} p_{\frac{3}{2}} \dots p_{\frac{(r - 1)}{2}} =$

The algebra $A_{k, r, 1} (n)$ is generated by $s_{1}, \dots, s_{k - 1}, p_{\frac{3}{2}}$ and $f_{r} .$

Is $C C A_{k} (r, p, n) = ℂ A_{k} (r, p, n) \times ℤ / p ℤ ?$

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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