Paul Martin: Reference Section
: Projects...
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: