Paul Martin: Reference Section
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As the picture indicates, we want to make a new braid picture by twisting the indicated part in the middle through 180 degrees. Of course this will not change the braid, but it will now be written $b = M g_1 M^{1}$ where
$M=$
It will be evident from this that $M^2$ lies in the centre of the braid group. Thus it also lies in the centre of the group algebra; and also in the centre of any quotient algebra ...such as the Hecke algebra.
By Schur's Lemma it follows that we can use $M^2$ to study the blocks of the Hecke algebra, or indeed any further quotient such as the TL algebra.
Exercise: compute the action of $M^2$ on TL standard modules. (This is easy.)

Consider the Brauer algebra B_n as a subalgebra of the partition algebra. (See here for the partition algebra.) Consider in particular the diagram basis (certain pictures of partitions of a set of n+n elements drawn in a rectangular frame), and the algebra multiplication formulated in terms of a corresponding diagram juxtaposition. From this one sees that there is a subalgebra with basis the subset of noncrossing diagrams. This is the TL algebra  an algebra of considerable significance in many areas of mathematics and physics.
A handy way to think of the TL algebra is as a quotient of the Hecke algebra, which is in turn a quotient of the group algebra of the braid group.
Finally,... The blob algebra is a generalisation of the TL algebra as we now indicate.
Some nice looking recent papers on the blob algebra, indicating a lot of interesting open problems (!):