A basis of the degenerate affine BMW algebra 𝒲k

Arun Ram
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 𝒲k

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,
d= is a Brauer diagram on 7 dots. (4.1)
Number the vertices of the top row, left to right, with 1,2,k and the vertices in the bottom row, left to right, with 1,2, ,k so that the diagram above can be written d= (13) (21) (45) (66) (74) (27) (35) . The Brauer algebra is the vector space
𝒲1,k with basis Dk = {diagrams on  k  dots } , (4.2)
and product given by placing diagrams on top of each other and changing each closed loop to x. For example, if
d1= and d2=
d1 d2= =x (4.3)
The Brauer algebra is generated by
si= e1= , , 1 i k-1, i i+1 i i+1 (4.4)
and is a subalgebra of the degenerate affine BMW algebra 𝒲k. The Brauer algebra is also the quotient of 𝒲k by y1=0 and, hence, can be viewed as the degnerate cyclotomic BMW algebra 𝒲1,k (0).

Let 𝒲k be the degenerate affine BMW algebra and let 𝒲r,k (u1,, ur) be the degnerate cyclotomic BMW algebra as defined in (2.11), (2.12) and (2.19), respectively. For n1 ,,nk and a diagram d on k dots let d n1 ,, nk = yi1n1 ,, yin d y i+1 n+1 ,, yik nk , where, in the lexicographic ordering of the edges (i1, j1) ,, (ik, jk) of d, i1,, i are in the top row of d and i+1 ,, ik are in the bottom row of d. Let Dk be the set of diagrams on k dots as in (???).

(a)   If ?????????? and ??????????????? then { d n1 ,, nk | dDk , n1 , nk } is a C-basis of 𝒲k.
(b)   If ?????????????? holds and ???????????? then { d n1 ,,nk | dDk , 0n1 ,,nk r-1 } is a C-basis of 𝒲r,k (b1,, br) .

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 { d n1 ,, nk | dDk , n1,, nk } spans 𝒲k is that if (i,j) is a top-to-bottom edge in d, then

yid = dyj +(terms with fewer crossings) ,
and if (i,j) is a top-to-top edge in d then
yid = - yjd +(terms with fewer crossings) .
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 u1, ,ur and the parameters z1() . See [AMR] for specifics.


[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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