The center of the 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: 24 April 2011

The center of the affine BMW algebra $W_k$

The affine BMW algebra $W_k$ is an algebra over the commutative ring $C$ and the polynomial ring $C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ is a subalgebra of $W_k$. The symmetric group $S_k$ acts on $C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ by permuting the variables and the ring of symmetric functions is $${C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}=\{f\in C[y_1,\dots ,y_k\left]\right|wf=f,\text{for}w\in S_k\}.$$ A classical fact (see, for example [GV, Proposition 2.1]) is that the center of the affine Hecke algebra $H_k$ is $$Z\left({\mathscr{H}}_k\right)={C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}.$$ The following theorem gives an analogous characterization of the center of the degenerate affine BMW algebra.

The center of the degenerate affine BMW algebra $W_k$ is $$R_k=\{f\in {C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}|f\left(Y_1,Y_1^{-1},Y_3,\dots ,Y_k\right)=f(1,1,Y_3,\dots ,Y_k)\}.$$

Proof.
Step 1: $f\in W_k$ commutes with all $y_i$ $\iff$ $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$:
Assume $f\in W_k$ and write $$f=\sum c_d^{n_1,\dots ,n_k}{T}_d^{n_1,\dots ,n_k}$$ in terms of the basis in Theorem 3.1?????????. Let $d\in D_k$ with the maximal number of crossings such that $c_d^{n_1,\dots ,n_k}\ne 0$ and suppose there is an edge $(i,j)$ of $d$ such that $j\ne i\prime$. Then, by (4.22) and (4.23), $$\text{the coefficient of}{Y}_i{T}_d^{n_1,\dots ,n_k}\text{in}{Y}_{i}f\text{is}{c}_d^{n_1,\dots ,n_k}$$ and $$\text{the coefficient of}{Y}_i{T}_d^{n_1,\dots ,n_k}\text{in}f{Y}_i\text{is}0.$$ If $Y_{i}f=f{Y}_i$, it follows that there is no such edge, and so $d=1$ (and therefore $T_d=1$). Thus $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$. Conversely, if $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ then $Y_{i}f=f{Y}_i$.

Step 2. $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ commutes with all $T_i$ $\iff$ $f\in R_k$ :
Assume $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ and write

$f=\sum _{a,b\in \mathbb{Z}}{Y}_1^a{Y}_2^b{f}_{a,b}$,    where $f_{a,b}\in C[Y_3^{\pm 1},\dots ,Y_k^{\pm 1}]$.

Then $f(1,1,Y_3,\dots Y_k)=\sum _{a,b\in \mathbb{Z}}{f}_{a,b}$ and

$f(Y_1,Y_1^{-1},Y_3,\dots ,Y_k)=\sum _{a,b\in \mathbb{Z}}{Y}_1^{a-b}{f}_{a,b}=\sum _{\ell \in \mathbb{Z}}{Y}_1^{\ell }\left(\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}\right)$.

(4.25)

By direct computation using (3.31) and (3.33),

$T_1{Y}_1^a{Y}_2^b=Y_1^a{Y}_1^a{T}_1{Y}_2^{b-a}=s_1\left(Y_1^a{Y}_2^b\right)T_1+(q-q^{-1}){\displaystyle \frac{Y_1^a{Y}_2^b-s_1\left(Y_1^a{Y}_2^b\right)}{1-Y_1{Y}_2^{-1}}}+{ℰ}_{b-a}$,

where

${ℰ}_{\ell }=\{\begin{array}{ll}-(q-q^{-1}){\displaystyle \sum _{r=1}^{\ell }}{Y}_1^{\ell -r}{E}_1{Y}_1^{-r},& \text{if}\ell >0,\\ (q-q^{-1}){\displaystyle \sum _{r=1}^{-\ell }}{Y}_1^{\ell +r-1}{E}_1{Y}_1^{r-1},& \text{if}\ell <0,\\ 0,& \text{if}\ell =0.\end{array}$

It follows that

$T_{1}f=\left(s_{1}f\right)T_1+(q-q^{-1}){\displaystyle \frac{f-s_{1}f}{1-Y_1{Y}_2^{-1}}}+\sum _{\ell \in {\mathbb{Z}}_{\ne 0}}{ℰ}_{\ell }\left(\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}\right)$.

(4.26)

Thus if $f\left(Y_1,Y_1^{-1},Y_3,\dots ,Y_k\right)=f(1,1,Y_3,\dots ,Y_k)$ then, by (4.25),

$\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}=0,\text{for}\ell \ne 0$.

(4.27)

Hence, if $f\in {C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}$ and $f\left(Y_1,Y_1^{-1},Y_3,\dots ,Y_k\right)=f(1,1,Y_3,\dots ,Y_k)$ then $s_{1}f=f$ and (4.27) holds so that, by (4.26), $T_{1}f=f{T}_1$. Similarly, $f$ commutes with all $T_i$.
Conversely, if $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ and $T_{i}f=f{T}_i$ then

$s_{i}f=f$     and     $\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}=0,\text{for}\ell \ne 0$,

so that $f\in {C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}$ and $f\left(Y_1,Y_1^{-1},Y_3,\dots ,Y_k\right)=f(1,1,Y_3,\dots ,Y_k)$. □

The symmetric group $S_k$ acts on ${\mathbb{Z}}^k$ by permuting the factors. The ring

${C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}$     has basis     $\left\{m_{\lambda }\right|\lambda \in {\mathbb{Z}}^k\text{with}{\lambda }_1\ge {\lambda }_2\ge \cdots {\lambda }_k\},$

where

$m_{\lambda }=\sum _{\gamma \in S_k\lambda }{Y}_1^{{\gamma }_1}\cdots Y_k^{{\gamma }_k}$.

The elementary symmetric functions are

$e_r=m_{(1^r,0^{k-r})}\text{and}{e}_{-r}=m_{(0^{k-r},{(-1)}^r)},\text{for}r=0,1,\dots ,k,$

and the power sum symmetric functions are

$p_r=m_{(r,0^{k-1})}\text{and}{p}_{-r}=m_{(0^{k-1},-r)},\text{for}r\in {\mathbb{Z}}_{\ge 0}$.

The Newton identities (see [Mac, Ch. I (2.11′)]) say

$\ell e_{\ell }=\sum _{r=1}^{\ell }{(-1)}^{r-1}{p}_r{e}_{\ell -r}$     and     $\ell e_{-\ell }=\sum _{r=1}^{\ell }{(-1)}^{r-1}{p}_{-r}{e}_{-(\ell -r)}$,

where the second equation is obtained from the first by replacing $Y_i$ with $Y_i^{-1}$. For $\ell \in \mathbb{Z}$ and $\lambda =({\lambda }_1,\dots ,{\lambda }_k)\in {\mathbb{Z}}^k$,

$e_k^{\ell }{m}_{\lambda }=m_{\lambda +\left({\ell }^k\right)}$,     where  $\lambda +\left({\ell }^k\right)=({\lambda }_1+\ell ,\dots ,{\lambda }_k+\ell )$.

In particular,

$e_{-r}=e_k^{-1}{e}_{k-r},\text{for}r=0,1,\dots k$.

(4.29)

Define

$p_i^{+}=p_i+p_{-i},p_i^{-}=p_i-p_{-i},\text{for}i\in {\mathbb{Z}}_{>0}$.

The consequence of (4.29) and (4.28) is that

$\begin{array}{rl}{\mathbb{Q}[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}& =\mathbb{Q}[e_k^{\pm 1},e_1,\dots ,e_{k-1}]\\ & =\mathbb{Q}\left[e_k^{\pm 1}\right][e_1,e_2,\dots e_{\lfloor \frac{k}{2}\rfloor },e_k{e}_{-\lfloor \frac{k-1}{2}\rfloor },e_k{e}_{-2},e_k{e}_{-1}]\\ & =\mathbb{Q}\left[e_k^{\pm 1}\right][e_1,e_2,\dots e_{\lfloor \frac{k}{2}\rfloor },e_{-\lfloor \frac{k-1}{2}\rfloor },e_{-2},e_{-1}]\\ & =\mathbb{Q}\left[e_k^{\pm 1}\right][p_1,p_2,\dots p_{\lfloor \frac{k}{2}\rfloor },p_{-\lfloor \frac{k-1}{2}\rfloor },p_{-2},p_{-1}]\\ & =\mathbb{Q}\left[e_k^{\pm 1}\right][p_1^{+},p_2^{+},\dots p_{\lfloor \frac{k}{2}\rfloor }^{+},p_{\lfloor \frac{k-1}{2}\rfloor }^{-},p_2^{-},p_1^{-}].\end{array}$

For $\nu \in {\mathbb{Z}}^k$ with ${\nu }_1\ge \cdots \ge {\nu }_{\ell }>0$ define

$p_{\nu }^{+}=p_{{\nu }_1}^{+}\cdots p_{{\nu }_{\ell }}^{+}\text{and}{p}_{\nu }^{-}=p_{{\nu }_1}^{-}\cdots p_{{\nu }_{\ell }}^{-}$.

Then

${C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}\text{has basis}\left\{e_k^{\ell }{p}_{\lambda }^{+}{p}_{\mu }^{-}\right|\ell \in \mathbb{Z},\ell \left(\lambda \right)\le \lfloor \frac{k}{2}\rfloor ,\ell \left(\mu \right)\le \lfloor \frac{k-1}{2}\rfloor \}$.

(4.29)

In analogy with (4.12) we expect that if $R_k$ is as in Theorem 4.6 then

$R_k=C\left[e_k^{\pm 1}\right][p_1^{-},p_2^{-},\dots ]$.

(4.30)

The left ideal of $W_2$ generated by $E_1$ is $C\left[Y_1^{\pm 1}\right]E_1$. This is an infinite dimensional (generically irreducible) $W_2$-module on which $Z\left(W_2\right)$ acts by constants. Thus, as noted by IS THIS IN RUI SI?????, it follows that $W_2$ is not finitely generated as a $Z\left(W_2\right)$-module.

Notes and references

This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV]. The characterisation of the center of the affine BMW algebra given here is analogous to that for the degenerate affine BMW algebra as found in (???). PUT A REMARK ABOUT the solution of [St. Exercise 7.7] as found on [St., p. 497] in THIS CONTEXT???

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.

[FJ+] B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, and Y. Takeyama, Symmetric polynomials vaishing on the diagonals shifted by roots of unity, Int. Math. Res. Notices 2003 no. 18, 1015-1034, arXiv:math/0209126. MR??????

[GV] I. Grojnowski and M. Vazirani, Strong multiplicity one theorems for affine Hecke algebras of type A, Transformations Groups 6 (2001), 143-155. arXiv:math/??????. MR??????

[LLT] A. Lascoux, B. Leclerc, J.-Y. Thibon, Green polynomials and Hall-Littlewood functions at roots of unity, Europ. J. Combinatorics 15 (1994), 173-180. MR??????

[Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995. ISBN: 0-19-853489-2 MR1354144

[Mo] A.O. Morris, On an algebra of symmetric functions, Quart. J. Math. 16 (1965), 53-64. MR??????

[Pr] P. Pragacz, Algegro-geometric applications of Schur S and Q polynomials in Topics in invraiant theory (Paris, 1989/1990), Lecture Notes in Math. 1478, Springer, Berlin (1991), 130-191. MR1180989

[St] R.P. Stanley, Enumerative Combinatorics Vol. 2 Cambridge Univ. Press 1999. ISBN: 0-521-56069-1; 0-521-78987-7 MR1676282

[To] B. Totaro, Towards a Schubert calculus for complex reflection groups, Math. Proc. Camb. Phil. Soc. 134 (2003), 83-93. http://www.dpmms.cam.ac.uk/~bt219/papers.html MR??????

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