The center of 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: 23 April 2011

The center of the degenerate affine BMW algebra ${𝒲}_k$

The degenerate affine BMW algebra ${𝒲}_k$ is an algebra over the commutative ring $C$ and the polynomial ring $C[y_1,\dots ,y_k]$ is a subalgebra of ${𝒲}_k$. The symmetric group $S_k$ acts on $C[y_1,\dots ,y_k]$ by permuting the variables and the ring of symmetric functions is $${C[y_1,\dots ,y_k]}^{S_k}=\{f\in C[y_1,\dots ,y_k\left]\right|wp=p,\text{for}w\in S_k\}.$$ A classical fact (see, for example [Kl, Theorem 3.3.1]) is that the center of the degenerate affine Hecke algebra ${\mathscr{H}}_k$ is $$Z\left({\mathscr{H}}_k\right)={C[y_1,\dots ,y_k]}^{S_k}.$$ 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 $${ℛ}_k=\{f\in {C[y_1,\dots ,y_k]}^{S_k}|f(y_1,-y_1,y_3,\dots ,y_k)=f(0,0,y_3,\dots ,y_k)\}.$$

Proof. Step 1: $f$ commutes with all $y_i$ $\iff$ $f\in C[y_1,\dots ,y_k]$:
Assume $f\in {𝒲}_k$ and write $$f=\sum c_d^{n_1,\dots ,n_k}{d}^{n_1,\dots ,n_k}$$ 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 (3.5) and (3.6) (ELSEWHERE) $$\text{the coefficient of}{y}_i{d}^{n_1,\dots ,n_k}\text{in}{y}_{i}f\text{is}{c}_d^{n_1,\dots ,n_k}$$ and $$\text{the coefficient of}{y}_i{d}^{n_1,\dots ,n_k}\text{in}f{y}_i\text{is}0.$$ If $y_{i}f=f{y}_i$, it follows that there is no such edge, and so $d=1$. Thus $f\in C[y_1,\dots ,y_k]$. Conversely, if $f\in C[y_1,\dots ,y_k]$ then $y_{i}f=f{y}_i$.

Step 2. $f\in C[y_1,\dots ,y_k]$ commutes with all $t_{s_i}$ $\iff$ $f\in {ℛ}_k$ :
Assume $f\in C[y_1,\dots ,y_k]$ and write

$f=\sum _{a,b\in {\mathbb{Z}}_{\ge 0}}{y}_1^a{y}_2^b{f}_{a,b}$,    where $f_{a,b}\in C[y_3,\dots ,y_k]$.

Then $f(0,0,y_3,\dots y_k)=\sum _{a,b\in {\mathbb{Z}}_{\ge 0}}{f}_{a,b}$ and

$f(y_1,-y_1,y_3,\dots ,y_k)=\sum _{a,b\in {\mathbb{Z}}_{\ge 0}}{(-1)}^b{y}_1^{a+b}{f}_{a,b}=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{y}_1^{\ell }\left(\sum _{b=0}^{\ell }{(-1)}^b{f}_{\ell -b,b}\right)$.

(4.9)

By direct computation using (3.11) and (3.12),

$t_{s_1}{y}_1^a{y}_2^b=s_1\left(y_1^a{y}_2^b\right)t_{s_1}-{\displaystyle \frac{y_1^a{y}_2^b-s_1\left(y_1^a{y}_2^b\right)}{y_1-y_2}}+{(-1)}^a\sum _{r=1}^{a+b}{(-1)}^r{y}_1^{a+b-r}{e}_1{y}_1^{r-1}$,

and it follows that

$t_{s_1}f=\left(s_{1}f\right)t_{s_1}-{\displaystyle \frac{f-s_{1}f}{y_1-y_2}}+\sum _{\ell \in {\mathbb{Z}}_{>0}}\left(\left(\sum _{r=1}^{\ell }{(-1)}^r{y}_1^{\ell -r}{e}_1{y}_1^{r-1}\right)\left(\sum _{b=0}^{\ell }{(-1)}^{\ell -b}{f}_{\ell -b,b}\right)\right)$.

(4.10)

Hence, if $f\in {C[y_1,\dots ,y_k]}^{S_k}$ and $f(y_1,-y_1,y_3,\dots ,y_k)=f(0,0,y_3,\dots ,y_k)$ then $s_{1}f=f$ and, by (4.9), (4.11) holds so that, by (4.10), $t_{s_1}f=f{t}_{s_1}$. Similarly, $f$ commutes with all $t_{s_i}$.
Conversely, if $f\in C[y_1,\dots ,y_k]$ and $t_{s_i}f=f{t}_{s_i}$ then

$s_{i}f=f$     and     $\sum _{b=0}^{\ell }{(-1)}^{\ell -b}{f}_{\ell -b,b}=0$,   for  $\ell \ne 0$,

so that $f\in {C[y_1,\dots ,y_k]}^{S_k}$ and $f(y_1,-y_1,y_3,\dots ,y_k)=f(0,0,y_3,\dots ,y_k)$. □

The power sum symmetric functions $p_i$, $i\in {\mathbb{Z}}_{>0}$, are given by

$p_i=y_1^i+y_2^i+\cdots +y_k^i$.

The Hall-Littlewood polynomials (see [Mac, Ch. III (2.1)]) are given by

$P_{\lambda }(y;t)=P_{\lambda }(y_1,\dots ,y_k;t)={\displaystyle \frac{1}{v_{\lambda }\left(t\right)}}\sum _{w\in S_k}w(y_1^{{\lambda }_1}\cdots y_k^{{\lambda }_k}\prod _{1\le i<j\le k}{\displaystyle \frac{x_i-t{x}_j}{x_i-x_j}})$,

where $v_{\lambda }\left(t\right)$ is a normalizing constant (a polynomial in $t$) so that the coefficient of $y_1^{{\lambda }_1}\cdots y_k^{{\lambda }_k}$ in $P_{\lambda }(y;t)$ is equal to 1. The Schur Q-functions (see [Mac, Ch. III (8.7)]) are

$Q_{\lambda }=\{\begin{array}{ll}0,& \text{if}\lambda is\; not\; strict,\\ 2^{\ell \left(\lambda \right)}{P}_{\lambda }(y;-1),& \text{if}\lambda is\; strict,\end{array}$

where $\ell \left(\lambda \right)$ is the number of parts of $\lambda$ and the partition $\lambda$ is strict if all its parts are distinct. Let ${ℛ}_k$ be as in Theorem 4.2. Then (see [Naz, Cor. 4.10], [Pr, Theorem 2.11(Q)] and [Mac, Ch. III §8])

${ℛ}_k=C[p_1,p_3,p_5,\dots ]=C\text{-span}\left\{Q_{\lambda }\right|\lambda \text{is strict}\}$.

(4.12)

More generally, let $r\in {\mathbb{Z}}_{>0}$ and let $\zeta$ be a primitive $r$th root of unity. Define

${ℛ}_{r,k}=\{f\in {\mathbb{Z}\left[\zeta \right][y_1,\dots ,y_k]}^{S_k}|f(y_1,\zeta y_1,\dots {\zeta }^{r-1}{y}_1,y_{r+1},\dots ,y_k)=f(0,0,\dots 0,y_{r+1},\dots ,y_k)\}$.

Then

${ℛ}_{r,k}{\otimes }_{\mathbb{Z}\left[\zeta \right]}\mathbb{Q}\left(\zeta \right)=\mathbb{Q}\left(\zeta \right)\left[p_i\right|i\ne 0\mathrm{mod}r]$,

(4.13)

and ${ℛ}_{r,k}$ has $\mathbb{Z}\left[\zeta \right]$-bases

$\left\{P_{\lambda }\right(y;\zeta \left)\right|m_i\left(\lambda \right)<r\text{and}{\lambda }_1\le k\left\}\text{and}\right\{q_{\lambda }(y;\zeta )\left|m_i\right(\lambda )<r\text{and}{\lambda }_1\le k\}$,

(4.14)

where $m_i\left(\lambda \right)$ is the number of parts of size $i$ in $\lambda$. The ring ${ℛ}_{r,k}$ is studied in [Mo], [LLT], [Mac, Ch. II Ex. 5.7 and Ex. 7.7], [To], [FJ+], and others. The proofs of (4.13) and (4.14) follow from [Mac, Ch. III Ex. 7.7.], [To, Lemma 2.2 and following remarks] and the arguments in the proofs of [FJ+, Lemma 3.2 and Proposition 3.5].

Notes and references

This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV]. Nazarov give a characterization of $Z\left({𝒲}_k\right)$ in [Naz, Cor 4.10]. the solution of [St. Exercise 7.7] as found on [St., p. 497],

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??????

[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??????

page history