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### The Hecke algebras and alcove geometry (this page is a work in progress!)

We assume you are familiar with the braid group (see e.g. [Martin91] Ch.5). Here is a braid drawn from top to bottom, separated nominally into three parts, as it were $b = 1 \times g_{n-1} \times 1$: = 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.)

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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 non-crossing 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 (!):

Some original refs: