Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last updates: 21 April 2011
A basis of the degenerate affine BMW algebra
A diagram on dots is a graph with
dots in the top row, dots in the bottom row
and edges pairing the dots. For example,
is a Brauer diagram on 7 dots.
Number the vertices of the top row, left to right, with
and the vertices in the bottom row, left to right, with
so that the diagram above can be written
The Brauer algebra is the vector space
and product given by placing diagrams on top of each other and changing each closed loop to
. For example, if
The Brauer algebra is generated by
and is a subalgebra of the degenerate affine BMW algebra
The Brauer algebra is also the quotient of by
and, hence, can be viewed as the degnerate cyclotomic BMW algebra
Let be the degenerate affine BMW algebra and let
be the degnerate cyclotomic BMW algebra as defined in (2.11), (2.12) and (2.19), respectively.
For and a diagram d on k dots let
where, in the lexicographic ordering of the edges
are in the top row of
are in the bottom row of .
Let be the set of diagrams on dots
as in (???).
If ?????????? and ???????????????
is a -basis of
If ?????????????? holds and ????????????
is a -basis of
Part (a) of the theorem is [Naz, Theorem 4.6] (see also [AMR, Theorem 2.12])
and part (b) is [AMR, Prop. 2.15 and Theorem 5.5]. We refer to these references for the proof,
remarking only that one key point in showing that
is that if is a top-to-bottom edge in
and if is a top-to-top edge in
This is illustrated in the affine case in (3.14).
Notes and references
This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV].
The basis theorem for the degenerate cyclotomic BMW algebra may not be quite correct
in its statement above and may require some conditions relating the parameters
and the parameters
. See [AMR] for specifics.