The degenerate affine BMW algebra Wk and its quotients

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: 11 April 2011

The degenerate affine BMW algebra $𝒲_{k}$ and its quotients

Let $C$ be a commutative ring and let $𝔹_{k}$ be the degenerate affine braid algebra over $C$ as defined in The degenerate affine braid algebra. Define $e_{i}$ in the degenerate affine braid algebra by

t_{s_{i}} y_{i} = y_{i + 1} t_{s_{i}} - (1 - e_{i}), for i = 1, \dots, k - 1,

(edb)

so that, with

t_{i, i + 1}

as in (gba4),

t_{i, i + 1} t_{s_{i}} = 1 - e_{i} .

(edb′)

Fix constants

ϵ = \pm 1 and z_{0}^{(ℓ)} \in C

, for

ℓ \in ℤ_{\geq 0}

The degenerate affine Birman-Wenzl-Murakami (BMW) algebra

𝒲_{k}

(with parameters

ϵ

and

z_{0}^{(ℓ)}

) is the quotient of the degenerate affine braid algebra

𝔹_{k}

by the relations

e_{i} t_{s_{i}} = t_{s_{i}} e_{i} = ϵ e_{i}, e_{i} t_{s_{i - 1}} e_{i} = e_{i} t_{s_{i + 1}} e_{i} = ϵ e_{i},

(dbw1)

e_{1} y_{1}^{𝓁} e_{1} = z_{1}^{(𝓁)} e_{1}, e_{i} (y_{i} + y_{i + 1}) = 0 = (y_{i} + y_{i + 1}) e_{i} .

(dbw2)

Conjugating (edb) by $t_{s_{i}}$ and using the first relation in (dbw1) gives

y_{i} t_{s_{i}} = t_{s_{i}} y_{i + 1} - (1 - e_{i}) .

(dbw3)

Then, by (edb′) and (gba4),

t_{i, i + 1} = t_{s_{i}} - ϵ e_{i}, and e_{i + 1} = t_{s_{i}} t_{s_{i + 1}} e_{i} t_{s_{i + 1}} t_{s_{i}} .

(dbw4)

Multiply the second relation in (dbw4) on the left and the right by

e_{i}

and then use the relations in (dbw1) to get

e_{i} e_{i + 1} e_{i} = e_{i} t_{s_{i}} t_{s_{i + 1}} e_{i} t_{s_{i + 1}} t_{s_{i}} t_{s_{i}} e_{i} = e_{i} t_{s_{i + 1}} e_{i} t_{s_{i + 1}} e_{i} = ϵ e_{i} t_{s_{i + 1}} e_{i} = e_{i},

so that

e_{i} e_{i \pm 1} e_{i} = e_{i} . Note that e_{i}^{2} = z_{1}^{(0)} e_{i}

(dbw5)

is a special case of the first identity in (dbw2). The relations

e_{i + 1} e_{i} = e_{i + 1} t_{s_{i}} t_{s_{i + 1}}, e_{i} e_{i + 1} = t_{s_{i + 1}} t_{s_{i}} e_{i + 1},

(dbw6)

t_{s_{i}} e_{i + 1} e_{i} = t_{s_{i + 1}} e_{i}, and e_{i + 1} e_{i} t_{s_{i + 1}} = e_{i + 1} t_{s_{i}}

(dbw7)

result from

\begin{array}{rcl} e_{i + 1} t_{s_{i}} t_{s_{i + 1}} & = & ϵ e_{i + 1} t_{s_{i}} e_{i + 1} t_{s_{i}} t_{s_{i + 1}} = e_{i + 1} t_{s_{i + 1}} t_{s_{i}} e_{i + 1} t_{s_{i}} t_{s_{i + 1}} = e_{i + 1} e_{i}, \\ t_{s_{i + 1}} t_{s_{i}} e_{i + 1} & = & ϵ t_{s_{i + 1}} t_{s_{i}} e_{i + 1} t_{s_{i}} e_{i + 1} = t_{s_{i + 1}} t_{s_{i}} e_{i + 1} t_{s_{i}} t_{s_{i + 1}} e_{i + 1} = e_{i} e_{i + 1}, \\ t_{s_{i}} e_{i + 1} e_{i} & = & ϵ t_{s_{i}} e_{i + 1} t_{s_{i}} e_{i} = ϵ t_{s_{i + 1}} e_{i} t_{s_{i + 1}} e_{i} = t_{s_{i + 1}} e_{i}, and \\ e_{i + 1} e_{i} t_{s_{i + 1}} & = & ϵ e_{i + 1} t_{s_{i + 1}} e_{i} t_{s_{i + 1}} = ϵ e_{i + 1} t_{s_{i}} e_{i + 1} t_{s_{i}} = e_{i + 1} t_{s_{i}} . \end{array}

A consequence (see (???)) of the defining relations of $𝒲_{k}$ is the equation

(z_{0} (- u) - (\frac{1}{2} + ϵ u)) (z_{0} (u) - (\frac{1}{2} - ϵ u)) e_{1} = (\frac{1}{2} - ϵ u) (\frac{1}{2} + ϵ u) e_{1},

where

z_{0} (u)

is the generating function

z_{0} (u) = \sum_{ℓ \in ℤ_{\geq 0}} z_{0}^{(ℓ)} u^{- ℓ} .

This means that, unless the parameters

z_{0}^{(ℓ)} are chosen carefully, it is likely that e_{1} = 0 in 𝒲_{k} .

From the point of view of the Schur-Weyl duality for the degenerate affine BMW algebra (see [AS] and [DRV]) the natural choice of base ring is the center of the enveloping algebra of the orthogonal or symplectic Lie algebra, which, by the Harish-Chandra isomorphism, is isomorphic to the subring of symmetric functions given by $C = {z \in {ℂ [h_{1}, \dots h_{r}]}^{S_{r}} | z (h_{1}, \dots h_{r}) = z (- h_{1}, h_{2}, \dots h_{r})},$ where the symmetric group $S_{r}$ acts by permuting the variables $h_{1}, \dots h_{r}$ . Here the constants $z_{0}^{(ℓ)} \in C$ are given explicitly, by setting the generating function

z_{0} (u)

equal, up to a normalization, to

\prod_{i = 1}^{r} \frac{(u + \frac{1}{2} + h_{i}) (u + \frac{1}{2} - h_{i})}{(u - \frac{1}{2} - h_{i}) (u - \frac{1}{2} + h_{i})} .

This choice of

C

and the

z_{0}^{(ℓ)}

are the universal admissible parameters for

𝒲_{k}

Quotients of $𝒲_{k}$

The degenerate affine Hecke algebra $ℋ_{k}$ is the quotient of $𝒲_{k}$ by the relations

e_{i} = 0, for i = 1, \dots, k - 1 .

(dah)

Fix

u_{1}, \dots, u_{r} \in ℂ .

The degenerate cyclotomic BMW algebra

𝒲_{r, k} (u_{1} \dots u_{r})

is the degenerate affine BMW algebra with the additional relation

(y_{1} - u_{1}) \dots (y_{1} - u_{r}) = 0

(cyc)

The degenerate cyclotomic Hecke algebra

ℋ_{r, k} (u_{1}, \dots, u_{r})

is the graded Hecke algebra with the additional relation (cyc).

A consequence of the relation (dah) in $𝒲_{r, k} (u_{1} \dots u_{r})$ is

(z_{0} (u) + u - \frac{1}{2}) e_{1} = (u - \frac{1}{2} {(- 1)}^{r}) (\prod_{i = 1}^{r} \frac{u + b_{i}}{u - b_{i}}) e_{1}

This equation makes the data of the values

b_{i}

almost equivalent to the data of the

z_{0}^{(ℓ)}

Notes and References

This section is based on forthcoming joint work with Z. Daugherty and R. Virk [DRV]. The definition of the degenerate affine BMW algebra here differs from the original definition of Nazarov [Naz] (used also in the paper [AMR]):

(a) Instead of fixing an infinite number of parameters $z_{1}^{(𝓁)}$ we choose parameters $h_{1}, \dots, h_{r}$ and define the $z_{1}^{(𝓁)}$ in terms of the $h_{i}$ .
(b) We add an extra parameter $ϵ$ .

The motivation for (b), the new parameter

ϵ

is that we want to handle the symplectic case in tandem with the orthogonal case (see Actions of Tantalizers). In [Naz], Nazarov considered only the orthogonal case. The motivation for (a) comes from the approach of [OR] where the degenerate affine Hecke algebra is acting on a tensor space, and the

z_{1}^{(𝓁)}

are naturally elements of the center of the enveloping algebra

U𝔤

which is in Schur-Weyl duality with the affine BMW algebra and the center of the enveloping algebra is, by the Harish-Chandra isomorphism, isomorphic to the ring of Weyl group symmetric polynomials in

h_{1}, \dots, h_{r}

, a basis of the Cartan subalgebra of

𝔤

. This new definition of the degenerate affine BMW algebra, provides the same finite dimensional representation theory as the algebra originally defined by Nazarov [Naz].

The factor $ϵ$ in (dbw1) is slightly unusual in the context of the Brauer algebra diagrams. For the precise conversion to the usual Brauer algebra see Degenerate BMW bases.

Bibliography

[AMR] S. Ariki, A. Mathas and H. Rui, Cyclotomic Nazarov Wenzl algebras, Nagoya Math. J. 182, (2006), 47-134. MR2235339 (2007d:20005)

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

[Naz] M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996), no. 3, 664--693. MR1398116 (97m:20057)

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$ , Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)

page history

The degenerate affine BMW algebra 𝒲k

The degenerate affine BMW algebra 𝒲k and its quotients

Quotients of 𝒲k

Notes and References

Bibliography

The degenerate affine BMW algebra $𝒲_{k}$

The degenerate affine BMW algebra $𝒲_{k}$ and its quotients

Quotients of $𝒲_{k}$