A basis of the degenerate affine BMW algebra ${𝒲}_k$

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 ${𝒲}_k$

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

${𝒲}_{1,k}\text{with basis}{D}_k=\left\{\text{diagrams on }k\text{ dots}\right\}$,

(4.2)

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 ${𝒲}_k$. The Brauer algebra is also the quotient of ${𝒲}_k$ by $y_1=0$ and, hence, can be viewed as the degnerate cyclotomic BMW algebra ${𝒲}_{1,k}\left(0\right)$.

Let ${𝒲}_k$ be the degenerate affine BMW algebra and let ${𝒲}_{r,k}(u_1,\dots ,u_r)$ 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 $\left\{d^{n_1,\dots ,n_k}\right|d\in D_k,n_1,\dots n_k\in \mathbb{Z}\}$ is a $C$-basis of ${𝒲}_k$.

(b)   If ?????????????? holds and ???????????? then $\left\{d^{n_1,\dots ,n_k}\right|d\in D_k,0\le n_1,\dots ,n_k\le r-1\}$ is a $C$-basis of ${𝒲}_{r,k}(b_1,\dots ,b_r)$.

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 $\left\{d^{n_1,\dots ,n_k}\right|d\in D_k,n_1,\dots ,n_k\in \mathbb{Z}\}$ spans ${𝒲}_k$ 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.

References

[AMR] S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov Wenzl algebras, Nagoya Math. J. 182, (2006), 47-134. arXiv:math/0506467, MR2235339

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

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