A basis of the degenerate affine BMW algebra
${\mathcal{W}}_{k}$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 21 April 2011

A basis of the degenerate affine BMW algebra
${\mathcal{W}}_{k}$

A diagram on $k$ dots is a graph with
$k$ dots in the top row, $k$ dots in the bottom row
and $k$ edges pairing the dots. For example,

is a Brauer diagram on 7 dots.

(4.1)

Number the vertices of the top row, left to right, with
$1,2,\dots k$
and the vertices in the bottom row, left to right, with
$1\prime ,2\prime ,\dots ,k\prime $
so that the diagram above can be written
$$d=\left(13\right)\left(21\prime \right)\left(45\right)\left(66\prime \right)\left(74\prime \right)\left(2\prime 7\prime \right)\left(3\prime 5\prime \right).$$
The Brauer algebra is the vector space

and product given by placing diagrams on top of each other and changing each closed loop to
$x$. For example, if

and

then

(4.3)

The Brauer algebra is generated by

(4.4)

and is a subalgebra of the degenerate affine BMW algebra
${\mathcal{W}}_{k}$.
The Brauer algebra is also the quotient of ${\mathcal{W}}_{k}$ by
${y}_{1}=0$
and, hence, can be viewed as the degnerate cyclotomic BMW algebra
${\mathcal{W}}_{1,k}\left(0\right)$.

Let ${\mathcal{W}}_{k}$ be the degenerate affine BMW algebra and let
${\mathcal{W}}_{r,k}({u}_{1},\dots ,{u}_{r})$
be the degnerate cyclotomic BMW algebra as defined in (2.11), (2.12) and (2.19), respectively.
For ${n}_{1},\dots ,{n}_{k}\in \mathbb{Z}$ and a diagram d on k dots let
$${d}^{{n}_{1},\dots ,{n}_{k}}={y}_{{i}_{1}}^{{n}_{1}},\dots ,{y}_{{i}_{\ell}}^{{n}_{\ell}}d{y}_{{i}_{\ell +1}}^{{n}_{\ell +1}},\dots ,{y}_{{i}_{k}}^{{n}_{k}},$$
where, in the lexicographic ordering of the edges
$({i}_{1},{j}_{1}),\dots ,({i}_{k},{j}_{k})$ of
$d$,
${i}_{1},\dots ,{i}_{\ell}$ are in the top row of $d$
and
${i}_{\ell +1},\dots ,{i}_{k}$ are in the bottom row of $d$.
Let ${D}_{k}$ be the set of diagrams on $k$ dots
as in (???).

(a)
If ?????????? and ???????????????
then
$\left\{{d}^{{n}_{1},\dots ,{n}_{k}}\phantom{\rule{.5em}{0ex}}\right|\phantom{\rule{.5em}{0ex}}d\in {D}_{k},\phantom{\rule{.2em}{0ex}}{n}_{1},\dots {n}_{k}\in \mathbb{Z}\}$ is a $C$-basis of
${\mathcal{W}}_{k}$.

(b)
If ?????????????? holds and ????????????
then
$\left\{{d}^{{n}_{1},\dots ,{n}_{k}}\phantom{\rule{.5em}{0ex}}\right|\phantom{\rule{.5em}{0ex}}d\in {D}_{k},\phantom{\rule{0.5em}{0ex}}0\le {n}_{1},\dots ,{n}_{k}\le r-1\}$ is a $C$-basis of
${\mathcal{W}}_{r,k}({b}_{1},\dots ,{b}_{r})$.

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
$\left\{{d}^{{n}_{1},\dots ,{n}_{k}}\phantom{\rule{.5em}{0ex}}\right|\phantom{\rule{.5em}{0ex}}d\in {D}_{k},\phantom{\rule{.2em}{0ex}}{n}_{1},\dots ,{n}_{k}\in \mathbb{Z}\}$
spans ${\mathcal{W}}_{k}$
is that if $(i,j)$ is a top-to-bottom edge in
$d$, then

${y}_{i}d=d{y}_{j}+\left(\text{terms with fewer crossings}\right)$,

and if $(i,j)$ is a top-to-top edge in
$d$ then

${y}_{i}d=-{y}_{j}d+\left(\text{terms with fewer crossings}\right)$.

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
${u}_{1},\dots ,{u}_{r}$
and the parameters
${z}_{1}^{\left(\ell \right)}$. See [AMR] for specifics.