Affine and degenerate affine BMW algebras: The center

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 21 October 2013

The center of the affine and degenerate affine BMW algebras

In this section, we identify the center of ${𝒲}_k$ and $W_k\text{.}$ Both centers arise as algebras of symmetric functions with a “cancellation property” (in the language of [Pra1180989]) or “wheel condition” (in the language of [FJM0209126]). In the degenerate case, $Z\left({𝒲}_k\right)$ is the ring of symmetric functions in $y_1,\dots ,y_k$ with the $Q\text{-cancellation}$ property of Pragacz. By [Pra1180989, Theorem 2.11(Q)], this is the same ring as the ring generated by the odd power sums, which is the way that Nazarov [Naz1996] identified $Z\left({𝒲}_k\right)\text{.}$

The cancellation property in the case of $W_k$ is analogous, exhibiting the center of the affine BMW algebra $Z\left(W_k\right)$ as a subalgebra of the ring of symmetric Laurent polynomials. At the end of this section, in an attempt to make the theory for the affine BMW algebra completely analogous to that for the degenerate affine BMW algebra, we have formulated an alternate description of $Z\left(W_{\lambda }\right)$ as a ring generated by “negative” power sums.

A basis of ${𝒲}_k$

A (Brauer) 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, $$\begin{array}{cc}d=\begin{array}{c} \end{array}\text{is a Brauer diagram on 7 dots.}& \text{(4.1)}\end{array}$$ 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 in (4.1) can be written $$d=\left(13\right)\left(21\prime \right)\left(45\right)\left(66\prime \right)\left(74\prime \right)\left(2\prime 7\prime \right)\left(3\prime 5\prime \right)\text{.}$$ The Brauer algebra is the vector space $$\begin{array}{cc}{𝒲}_{1,k}\text{with basis}{D}_k=\left\{\text{diagrams on}\hspace{0.17em}k\hspace{0.17em}\text{dots}\right\},& \text{(4.2)}\end{array}$$ and product given by stacking diagrams and changing each closed loop to $x\text{.}$ For example, $$\begin{array}{c}\text{if}{d}_1=\begin{array}{c} \end{array}\text{and}{d}_2=\begin{array}{c} \end{array}\text{then}\\ \begin{array}{cc}{d}_1{d}_2=\begin{array}{c} \end{array}=x\begin{array}{c} \end{array}\text{.}& \text{(4.3)}\end{array}\end{array}$$ The Brauer algebra is generated by $$\begin{array}{cc}{e}_i=\begin{array}{c} i i+1 ⋯ ⋯ \end{array},s_i=\begin{array}{c} i i+1 ⋯ ⋯ \end{array},1\le i\le k-1\text{.}& \text{(4.4)}\end{array}$$ Setting $$x=z_0^{\left(0\right)}\text{and}{s}_i=\epsilon t_{s_i}$$ realizes the Brauer algebra as a subalgebra of the degenerate affine BMW algebra ${𝒲}_k\text{.}$ The Brauer algebra is also the quotient of ${𝒲}_k$ by $y_1=0$ and, hence, can be viewed as the degenerate cyclotomic BMW algebra ${𝒲}_{1,k}\left(0\right)\text{.}$

Let ${𝒲}_k$ be the degenerate affine BMW algebra and let ${𝒲}_{r,k}(b_1,\dots ,b_r)$ be the degenerate cyclotomic BMW algebra as defined in (2.23)-(2.24) and (2.31), respectively. For $n_1,\dots ,n_k\in {\mathbb{Z}}_{\ge 0}$ and a diagram $d$ on $k$ dots let $$d^{n_1,\dots ,n_k}=y_{i_1}^{n_1}\cdots y_{i_{\ell }}^{n_{\ell }}d{y}_{i_{\ell +1}}^{n_{\ell +1}}\cdots y_{i_k}^{n_k},$$ 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 $d\text{.}$ Let $D_k$ be the set of diagrams on $k$ dots, as in (4.2).

(a)	If ${\kappa }_0,{\kappa }_1\in C$ and $$\begin{array}{cc}(z_0(-u)-(\frac{1}{2}+\epsilon u))(z_0\left(u\right)-(\frac{1}{2}-\epsilon u))=(\frac{1}{2}-\epsilon u)(\frac{1}{2}+\epsilon u)& \text{(4.5)}\end{array}$$ then $\left\{d^{n_1,\dots ,n_k}\hspace{0.17em}\right|\hspace{0.17em}d\in D_k,\hspace{0.17em}{n}_1,\dots ,n_k\in {\mathbb{Z}}_{\ge 0}\}$ is a $C\text{-basis}$ of ${𝒲}_k\text{.}$
(b)	If ${\kappa }_0,{\kappa }_1\in C,$ (4.5) holds, and $$\begin{array}{cc}(z_0\left(u\right)+u-\frac{1}{2})=(u-\frac{1}{2}{(-1)}^r)\left(\prod _{i=1}^r\frac{u+b_i}{u-b_i}\right)& \text{(4.6)}\end{array}$$ then $\left\{d^{n_1,\dots ,n_k}\hspace{0.17em}\right|\hspace{0.17em}d\in D_k,\hspace{0.17em}0\le n_1,\dots ,n_k\le r-1\}$ is a $C\text{-basis}$ of ${𝒲}_{r,k}(b_1,\dots ,b_r)\text{.}$

Part (a) of Theorem 4.1 is [Naz1996, Theorem 4.6] (see also [AMR0506467, Theorem 2.12]) and part (b) is [AMR0506467, Prop. 2.15 and Theorem 5.5]. We refer to these references for the proof, remarking only that one key point in showing that $\left\{d^{n_1,\dots ,n_k}\hspace{0.17em}\right|\hspace{0.17em}d\in D_k,n_1,\dots ,n_k\in {\mathbb{Z}}_{\ge 0}\}$ spans ${𝒲}_k$ is that if $(i,j)$ is a top-to-bottom edge in $d,$ then $$\begin{array}{cc}{y}_{i}d=d{y}_j+\text{(terms with fewer crossings),}& \text{(4.7)}\end{array}$$ and if $(i,j)$ is a top-to-top edge in $d$ then $$\begin{array}{cc}{y}_{i}d=-y_{j}d+\text{(terms with fewer crossings).}& \text{(4.8)}\end{array}$$ This is illustrated in the affine case in (4.24).

The center of ${𝒲}_k$

The degenerate affine BMW algebra is the algebra ${𝒲}_k$ over $C$ defined in Section 2.2 and the polynomial ring $C[y_1,\dots ,y_k]$ is a subalgebra of ${𝒲}_k$ (see Remark 2.5). The symmetric group $S_k$ acts on $C[y_1,\dots ,y_k]$ by permuting the variables and the ring of symmetric functions is $$C{[y_1,\dots ,y_k]}^{S_k}=\{f\in C[y_1,\dots ,y_k]\hspace{0.17em}|\hspace{0.17em}wf=f\text{, for}\hspace{0.17em}w\in S_k\}\text{.}$$ A classical fact (see, for example, [Kle2165457, Theorem 3.3.1]) is that the center of the degenerate affine Hecke algebra ${\mathscr{H}}_k$ is $$Z\left({\mathscr{H}}_k\right)=C{[y_1,\dots ,y_k]}^{S_k}\text{.}$$ Theorem 4.2 gives an analogous characterization of the center of the degenerate affine BMW algebra.

The center of the degenerate affine BMW algebra ${𝒲}_k$ is $${ℛ}_k=\{f\in C{[y_1,\dots ,y_k]}^{S_k}\hspace{0.17em}|\hspace{0.17em}f(y_1,-y_1,y_3,\dots ,y_k)=f(0,0,y_3,\dots ,y_k)\}\text{.}$$

		Proof.
	Step 1: $f\in {𝒲}_k$ commutes with all $y_i\iff f\in C[y_1,\dots ,y_k]\text{:}$ 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 4.1. Let $d\in D_k$ with the maximal number of crossings such that $c_d^{n_1,\dots ,n_k}\ne 0$ and, using the notation before (4.2), suppose there is an edge $(i,j)$ of $d$ such that $j\ne i\prime \text{.}$ Then, by (4.7) and (4.8), $$\text{the coefficient of}{y}_i{d}^{n_1,\dots ,n_k}\text{in}{y}_{i}f\text{is}{c}_d^{n_1,\dots ,n_k}$$ and $$\text{the coefficient of}{y}_i{d}^{n_1,\dots ,n_k}\text{in}f{y}_i\text{is}0\text{.}$$ If $y_{i}f=f{y}_i,$ it follows that there is no such edge, and so $d=1\text{.}$ Thus $f\in C[y_1,\dots ,y_k]\text{.}$ Conversely, if $f\in C[y_1,\dots ,y_k]$ then $y_{i}f=f{y}_i\text{.}$ Step 2: $f\in C[y_1,\dots ,y_k]$ commutes with all $t_{s_i}\iff f\in {ℛ}_k\text{:}$ Assume $f\in C[y_1,\dots ,y_k]$ and write $$f=\sum _{a,b\in {\mathbb{Z}}_{\ge 0}}{y}_1^a{y}_2^b{f}_{a,b},\text{where}\hspace{0.17em}{f}_{a,b}\in C[y_3,\dots ,y_k]\text{.}$$ Then $f(0,0,y_3,\dots ,y_k)=\sum _{a,b\in {\mathbb{Z}}_{\ge 0}}{f}_{a,b}$ and $$\begin{array}{cc}f(y_1,-y_1,y_3,\dots ,y_k)=\sum _{a,b\in {\mathbb{Z}}_{\ge 0}}{(-1)}^b{y}_1^{a+b}{f}_{a,b}=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{y}_1^{\ell }\left(\sum _{b=0}^{\ell }{(-1)}^b{f}_{\ell -b,b}\right)\text{.}& \text{(4.9)}\end{array}$$ By direct computation using (3.10) and (3.11), $$t_{s_1}{y}_1^a{y}_2^b=s_1\left(y_1^a{y}_1^b\right)t_{s_1}-\frac{y_1^a{y}_2^b-s_1\left(y_1^a{y}_2^b\right)}{y_1-y_2}+{(-1)}^a\sum _{r=1}^{a+b}{(-1)}^r{y}_1^{a+b-r}{e}_1{y}_1^{r-1},$$ and it follows that $$\begin{array}{cc}{t}_{s_1}f=\left(s_{1}f\right)t_{s_1}-\frac{f-s_{1}f}{y_1-y_2}+\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\left(\left(\sum _{r=1}^{\ell }{(-1)}^r{y}_1^{\ell -r}{e}_1{y}_1^{r-1}\right)\left(\sum _{b=0}^{\ell }{(-1)}^{\ell -b}{f}_{\ell -b,b}\right)\right)\text{.}& \text{(4.10)}\end{array}$$ Thus, if $f(y_1,-y_1,y_3,\dots ,y_k)=f(0,0,y_3,\dots ,y_k),$ then $$\begin{array}{cc}\sum _{b=0}^{\ell }{(-1)}^b{f}_{\ell -b,b}=0,\text{for}\hspace{0.17em}\ell \ne 0\text{.}& \text{(4.11)}\end{array}$$ 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), $t_{s_1}f=f{t}_{s_1}\text{.}$ Similarly, $f$ commutes with all $t_{s_i}\text{.}$ Conversely, if $f\in C[y_1,\dots ,y_k]$ and $t_{s_i}f=f{t}_{s_i}$ then $$s_{i}f=f\text{and}\sum _{b=0}^{\ell }{(-1)}^{\ell -b}{f}_{\ell -b,b}=0,\text{for}\hspace{0.17em}\ell \ne 0,$$ so that $f\in C{[y_1,\dots ,y_k]}^{S_k}$ and $f(y_1,-y_1,y_3,\dots ,y_k)=f(0,0,y_3,\dots ,y_k)\text{.}$ $\square$

It follows from (2.21) that ${ℛ}_k=Z\left({𝒲}_k\right)\text{.}$

The power sum symmetric functions $p_i$ are given by $$p_i=y_1^i+y_2^i+\cdots +y_k^i,\text{for}\hspace{0.17em}i\in {\mathbb{Z}}_{\ge 0}\text{.}$$ The Hall-Littlewood polynomials (see [Mac1354144, Ch. III (2.1)]) are given by $$P_{\lambda }(y;t)=P_{\lambda }(y_1,\dots ,y_k;t)=\frac{1}{v_{\lambda }\left(t\right)}\sum _{w\in S_k}w\left(y_1^{{\lambda }_1}\cdots y_k^{{\lambda }_k}\prod _{1\le i<j\le k}\frac{x_i-t{x}_j}{x_i-x_j}\right),$$ where $v_{\lambda }\left(t\right)$ is a normalizing constant (a polynomial in $t\text{)}$ 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 $Q\text{-functions}$ (see [Mac1354144, Ch. III (8.7)]) are $$Q_{\lambda }=\{\begin{array}{cc}0,& \text{if}\hspace{0.17em}\lambda \hspace{0.17em}\text{is not strict,}\\ 2^{\ell \left(\lambda \right)}{P}_{\lambda }(y;-1),& \text{if}\hspace{0.17em}\lambda \hspace{0.17em}\text{is strict,}\end{array}$$ where $\ell \left(\lambda \right)$ is the number of (nonzero) parts of $\lambda$ and the partition $\lambda$ is strict if all its (nonzero) parts are distinct. Let ${ℛ}_k$ be as in Theorem 4.2. Then (see [Naz1996, Cor. 4.10], [Pra1180989, Theorem 2.11(Q)] and [Mac1354144, Ch. III §8]) $$\begin{array}{cc}{ℛ}_k=C[p_1,p_3,p_5,\dots ]=C\text{-span}\left\{Q_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \hspace{0.17em}\text{is strict}\}\text{.}& \text{(4.12)}\end{array}$$ More generally, let $r\in {\mathbb{Z}}_{>0}$ and let $\zeta$ be a primitive $r\text{th}$ root of unity. Define $${ℛ}_{r,k}\{f\in \mathbb{Z}\left[\zeta \right]{[y_1,\dots ,y_k]}^{S_k}\hspace{0.17em}|\hspace{0.17em}f(y_1,\zeta y_1,\dots ,{\zeta }^{r-1}{y}_1,y_{r+1},\dots ,y_k)=f(0,0,\dots ,0,y_{r+1},\dots ,y_k)\}\text{.}$$ Then $$\begin{array}{cc}{ℛ}_{r,k}{\otimes }_{\mathbb{Z}\left[\zeta \right]}\mathbb{Q}\left(\zeta \right)=\mathbb{Q}\left(\zeta \right)\left[p_i\hspace{0.17em}\right|\hspace{0.17em}i\ne 0\hspace{0.17em}\text{mod}\hspace{0.17em}r],& \text{(4.13)}\end{array}$$ and $$\begin{array}{cc}{ℛ}_{r,k}\hspace{0.17em}\text{has}\hspace{0.17em}\mathbb{Z}\left[\zeta \right]\text{-basis}\left\{P_{\lambda }(y;\zeta )\hspace{0.17em}\right|\hspace{0.17em}{m}_i\left(\lambda \right)<r\hspace{0.17em}\text{and}\hspace{0.17em}{\lambda }_1\le k\},& \text{(4.14)}\end{array}$$ where $m_i\left(\lambda \right)$ is the number parts of size $i$ in $\lambda \text{.}$ The ring ${ℛ}_{r,k}$ is studied in [Mor0246983], [LLT1261063], [Mac1354144, Ch. III Ex. 5.7 and Ex. 7.7], [Tot1937794], [FJM0209126], and others. The proofs of (4.13) and (4.14) follow from [Mac1354144, Ch. III Ex. 7.7], [Tot1937794, Lemma 2.2 and following remarks] and the arguments in the proofs of [FJM0209126, Lemma 3.2 and Proposition 3.5].

The left ideal of ${𝒲}_2$ generated by $e_1$ is $C\left[y_1\right]e_1\text{.}$ This is an infinite dimensional (generically irreducible) ${𝒲}_2\text{-module}$ on which $Z\left({𝒲}_2\right)$ acts by constants. Thus, as noted by [AMR0506467, par. before Ex. 2.17], it follows that ${𝒲}_2$ is not finitely generated as a $Z\left({𝒲}_2\right)\text{-module.}$

A basis of $W_k$

An affine tangle has $k$ strands and a flagpole just as in the case of an affine braid, but there is no restriction that a strand must connect an upper vertex to a lower vertex. Let $X^{{\epsilon }_1}$ and $T_i$ be the affine braids given in (2.37) and let $$\begin{array}{cc}{E}_i=\begin{array}{c} \end{array}\text{.}& \text{(4.15)}\end{array}$$ Goodman and Hauchschild [GHa0411155, Cor. 6.14(b)] have shown that the affine BMW algebra $W_k$ is the algebra of linear combinations of tangles generated by $X^{{\epsilon }_1},T_1\text{.}\dots ,T_{k-1},E_1,\dots E_{k-1}$ and the relations (2.42), (2.43) and (2.45) expressed in the form $$\begin{array}{cc}\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})& \text{(4.16)}\\ \begin{array}{c} \end{array}=z\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=z^{-1}\begin{array}{c} \end{array}& \text{(4.17)}\\ \ell \hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}=Z_0^{\left(\ell \right)}\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=z^{-1}·\begin{array}{c} \end{array}& \text{(4.18)}\\ \begin{array}{c} \end{array}=\frac{z-z^{-1}}{q-q^{-1}}+1=Z_0^{\left(0\right)}\text{.}& \text{(4.19)}\end{array}$$

Let $W_k$ be the affine BMW algebra and let $W_{r,k}(b_1,\dots ,b_r)$ be the cyclotomic BMW algebra as defined in Section 2.4. Let $d\in D_k$ be a Brauer diagram, where $D_k$ is as in (4.2). Choose a minimal length expression of $d$ as a product of $e_1,\dots ,e_{k-1},s_1,\dots ,s_{k-1},$ $$d=a_1\cdots a_{\ell },a_i\in \{e_1,\dots ,e_{k-1},s_1,\dots ,s_{k-1}\},$$ such that the number of $s_i$ in this product is the number of crossings in $d\text{.}$ For each $a_i$ which is in $\{s_1,\dots ,s_{k-1}\}$ fix a choice of sign ${\epsilon }_j=\pm 1$ and set $$T_d=A_1\cdots A_{\ell },\text{where}{A}_j=\{\begin{array}{cc}{E}_i,& \text{if}\hspace{0.17em}{a}_j=e_i,\\ T_i^{{\epsilon }_j},& \text{if}\hspace{0.17em}{a}_j=s_i\text{.}\end{array}$$ For $n_1,\dots ,n_k\in \mathbb{Z}$ let $$T_d^{n_1,\dots ,n_k}=Y_{i_1}^{n_1}\cdots Y_{i_{\ell }}^{n_{\ell }}{T}_d{Y}_{i_{\ell +1}}^{n_{\ell +1}}\cdots Y_{i_k}^{n_k},$$ 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 $d\text{.}$

(a)	If $$\begin{array}{cc}(Z_0^{-}-\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1})(Z_0^{+}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1})=\frac{-(u^2-q^2)(u^2-q^{-2})}{{(u^2-1)}^2{(q-q^{-1})}^2}& \text{(4.20)}\end{array}$$ then $\left\{T_d^{n_1,\dots ,n_{\lambda }}\hspace{0.17em}\right|\hspace{0.17em}d\in D_k,\hspace{0.17em}{n}_1,\dots ,n_k\in \mathbb{Z}\}$ is a $C\text{-basis}$ of $W_k\text{.}$
(b)	If (4.20) holds and $$\begin{array}{cc}{Z}_0^{+}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1}=(\frac{z}{q-q^{-1}}+\frac{uz}{u^2-1})\prod _{j=1}^r\frac{u-b_j^{-1}}{u-b_j}& \text{(4.21)}\end{array}$$ then $\left\{T_d^{n_1,\dots ,n_k}\hspace{0.17em}\right|\hspace{0.17em}d\in D_k,\hspace{0.17em}0\le n_1,\dots ,n_k\le r-1\}$ is a $C\text{-basis}$ of $W_{r,k}(b_1,\dots ,b_r)\text{.}$

Part (a) of Theorem 4.4 is [GMo0612064, Theorem 2.25] and part (b) is [GMo0612064, Theorem 5.5] and [WYu0911.5284, Theorem 8.1]. We refer to these references for proof, remarking only that one key point in showing that $\left\{T_d^{n_1,\dots ,n_k}\hspace{0.17em}\right|\hspace{0.17em}d\in D_k,\hspace{0.17em}{n}_1,\dots ,n_k\in \mathbb{Z}\}$ spans $W_k$ is that if $(i,j)$ is a top-to-bottom edge in $d$ then $$\begin{array}{cc}{Y}_i{T}_d=T_d{Y}_j+\text{(terms with fewer crossings),}& \text{(4.22)}\end{array}$$ and, if $(i,j)$ is a top-to-top edge in $d$ then $$\begin{array}{cc}{Y}_i{T}_d=Y_j^{-1}{T}_d+\text{(terms with fewer crossings).}& \text{(4.23)}\end{array}$$ As an example, let $d=s_1{e}_3{s}_5{e}_2{e}_4{e}_1{s}_3{s}_5$ and choose ${\epsilon }_1={\epsilon }_3=-{\epsilon }_7={\epsilon }_8=1\text{.}$ Then $$d=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\text{and}{T}_d=\begin{array}{c} \end{array}$$ so that $T_d=T_1{E}_3{T}_5{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5$ and $T_d^{5,3,-2,0,3,0}=Y_1^5{Y}_2^3{Y}_3^{-2}{T}_d{Y}_1^3\text{.}$ Then, since $(1,6)$ is a horizontal edge in $d,$ (4.23) is illustrated by the computation $$\begin{array}{cc}\begin{array}{ccc}{Y}_6{T}_d& =& T_1{E}_3{Y}_6{T}_5{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5=T_1{E}_3(T_5{Y}_5+(q-q^{-1})Y_6(1-E_5))E_2{E}_4{E}_1{T}_3^{-1}{T}_5\\ & =& T_1{E}_3{T}_5{E}_2{Y}_5{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots =T_1{E}_3{T}_5{E}_2{Y}_4^{-1}{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots \\ & =& T_1{E}_3{Y}_4^{-1}{T}_5{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots =T_1{E}_3{Y}_3{T}_5{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots \\ & =& T_1{E}_3{T}_5{Y}_3{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots =T_1{E}_3{T}_5{Y}_2^{-1}{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots \\ & =& T_1{Y}_2^{-1}{E}_3{T}_5{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots =Y_1^{-1}{T}_1{E}_3{T}_5{E}_2{E}_4{E}_1{T}_3^{-1}{T}_5+\cdots ,\end{array}& \text{(4.24)}\end{array}$$ where $+\cdots$ is always a linear combination of terms with fewer crossings.

The center of $W_k$

The affine BMW algebra is the algebra $W_k$ over $C$ defined in Section 2.4 and the ring of Laurent polynomials $C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ is a subalgebra of $W_k$ (see Remark 2.10). 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 $$C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}=\{f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]\hspace{0.17em}|\hspace{0.17em}wf=f\text{, for}\hspace{0.17em}w\in S_k\}\text{.}$$ A classical fact (see, for example, [GVa1835669, Proposition 2.1]) is that the center of the affine Hecke algebra $H_k$ is $$Z\left(H_k\right)=C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}\text{.}$$ Theorem 4.5 is a characterization of the center of the affine BMW algebra.

The center of the affine BMW algebra $W_k$ is $$R_k=\{f\in C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}\hspace{0.17em}|\hspace{0.17em}f(Y_1,Y_1^{-1},Y_3,\dots ,Y_k)=f(1,1,Y_3,\dots ,Y_k)\}\text{.}$$

		Proof.
	Step 1: $f\in W_k$ commutes with all $Y_i\iff f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]\text{:}$ 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 4.4. Let $d\in D_k$ with the maximal number of crossings such that $c_d^{n_1,\dots ,n_k}\ne 0$ and, using the notation before (4.2), suppose there is an edge $(i,j)$ of $d$ such that $j\ne i\prime \text{.}$ 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 $$\text{the coefficient of}{Y}_i{T}_d^{n_1,\dots ,n_k}\text{in}f{Y}_i\text{is}0\text{.}$$ If $Y_{i}f=f{Y}_i$ it follows that there is no such edge, and so $d=1$ (and therefore $T_d=1\text{).}$ Thus $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]\text{.}$ Conversely, if $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}],$ then $Y_{i}f=f{Y}_i\text{.}$ Step 2: $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ commutes with all $T_i\iff f\in R_k\text{:}$ Assume $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ and write $$f=\sum _{a,b\in \mathbb{Z}}{Y}_1^a{Y}_2^b{f}_{a,b},\text{where}\hspace{0.17em}{f}_{a,b}\in C[Y_3^{\pm 1},\dots ,Y_k^{\pm 1}]\text{.}$$ Then $f(1,1,Y_3,\dots ,Y_k)=\sum _{a,b\in \mathbb{Z}}{f}_{a,b}$ and $$\begin{array}{cc}f(Y_1,Y_1^{-1},Y_3,\dots ,Y_k)=\sum _{a,b\in \mathbb{Z}}{Y}_1^{a-b}{f}_{a,b}=\sum _{\ell \in \mathbb{Z}}{Y}_1^{\ell }\left(\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}\right)\text{.}& \text{(4.25)}\end{array}$$ By direct computation using (3.30) and (3.32), $$T_1{Y}_1^a{Y}_2^b=Y_1^a{Y}_2^a{T}_1{Y}_2^{b-a}=s_1\left(Y_1^a{Y}_2^b\right)T_1+(q-q^{-1})\frac{Y_1^a{Y}_2^b-s_1\left(Y_1^a{Y}_2^b\right)}{1-Y_1{Y}_2^{-1}}+{ℰ}_{b-a},$$ where $${ℰ}_{\ell }=\{\begin{array}{cc}-(q-q^{-1})\sum _{r=1}^{\ell }{Y}_1^{\ell -r}{E}_1{Y}_1^{-r},& \text{if}\hspace{0.17em}\ell >0,\\ (q-q^{-1})\sum _{r=1}^{\ell }{Y}_1^{\ell +r-1}{E}_1{Y}_1^{r-1},& \text{if}\hspace{0.17em}\ell <0,\\ 0,& \text{if}\hspace{0.17em}\ell =0\text{.}\end{array}$$ It follows that $$\begin{array}{cc}{T}_{1}f=\left(s_{1}f\right)T_1+(q-q^{-1})\frac{f-s_{1}f}{1-Y_1{Y}_2^{-1}}+\sum _{\ell \in {\mathbb{Z}}_{\ne 0}}{ℰ}_{\ell }\left(\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}\right)\text{.}& \text{(4.26)}\end{array}$$ Thus, if $f(Y_1,Y_1^{-1},Y_3,\dots ,Y_k)=f(1,1,Y_3,\dots ,Y_k)$ then, by (4.25), $$\begin{array}{cc}\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}=0,\text{for}\hspace{0.17em}\ell \ne 0\text{.}& \text{(4.27)}\end{array}$$ Hence, if $f\in C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}$ and $f(Y_1,Y_1^{-1},Y_3,\dots ,Y_k)=f(1,1,Y_3,\dots ,Y_k)$ then $s_{1}f=f$ and (4.27) holds so that, by (4.26), $T_{1}f=f{T}_1\text{.}$ Similarly, $f$ commutes with all $T_i\text{.}$ Conversely, if $f\in C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ and $T_{i}f=f{T}_i$ then $$s_{i}f=f\text{and}\sum _{b\in \mathbb{Z}}{f}_{\ell +b,b}=0,\text{for}\hspace{0.17em}\ell \ne 0,$$ so that $f\in C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}$ and $f(Y_1,Y_1^{-1},Y_3,\dots ,Y_k)=f(1,1,Y_3,\dots ,Y_k)\text{.}$ It follows from (2.41) that $R_k=Z\left(W_k\right)\text{.}$ $\square$

The symmetric group $S_k$ acts on ${\mathbb{Z}}^k$ by permuting the factors. The ring $$C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}\text{has basis}\left\{m_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {\mathbb{Z}}^k\hspace{0.17em}\text{with}\hspace{0.17em}{\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_k\},$$ where $$m_{\lambda }=\sum _{\mu \in S_k\lambda }{Y}_1^{{\mu }_1}\cdots Y_k^{{\mu }_k}\text{.}$$ The elementary symmetric functions are $$e_r=m_{(1^r,0^{k-r})}\text{and}{e}_{-r}=m_{(0^{k-r},{(-1)}^r)},\text{for}\hspace{0.17em}r=0,1,\dots ,k,$$ and the power sum symmetric functions are $$p_r{m}_{(r,0^{k-1})}\text{and}{p}_{-r}=m_{(0^{k-1},-r)},\text{for}\hspace{0.17em}r\in {\mathbb{Z}}_{>0}\text{.}$$ The Newton identities (see [Mac1354144, Ch. I (2.11′)]) say $$\begin{array}{cc}\ell e_{\ell }=\sum _{r=1}^{\ell }{(-1)}^{r-1}{p}_r{e}_{\ell -r}\text{and}\ell e_{-\ell }=\sum _{r=1}^{\ell }{(-1)}^{r-1}{p}_{-r}{e}_{-(\ell -r)},& \text{(4.28)}\end{array}$$ where the second equation is obtained from the first by replacing $Y_i$ with $Y_i^{-1}\text{.}$ For $\ell \in \mathbb{Z}$ and $\lambda =({\lambda }_1,\dots ,{\lambda }_k)\in {\mathbb{Z}}^k,$ $$e_k^{\ell }{m}_{\lambda }=m_{\lambda +\left({\ell }^k\right)},\text{where}\lambda +\left({\ell }^k\right)=({\lambda }_1+\ell ,\dots ,{\lambda }_k+\ell )\text{.}$$ In particular, $$\begin{array}{cc}{e}_{-r}=e_k^{-1}{e}_{k-r},\text{for}\hspace{0.17em}r=0,\dots ,k\text{.}& \text{(4.29)}\end{array}$$ Define $$\begin{array}{cc}{p}_i^{+}=p_i+p_{-i}\text{and}{p}_i^{-}=p_i-p_{-i},\text{for}\hspace{0.17em}i\in {\mathbb{Z}}_{>0}\text{.}& \text{(4.30)}\end{array}$$ The consequence of (4.29) and (4.28) is that $$\begin{array}{ccc}ℂ{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}& =& ℂ[e_k^{\pm 1},e_1,\dots ,e_{k-1}]\\ & =& ℂ\left[e_k^{\pm 1}\right][e_1,e_2,\dots ,e_{\lfloor \frac{k}{2}\rfloor },e_k{e}_{-\lfloor \frac{k-1}{2}\rfloor },\dots ,e_k{e}_{-2},e_k{e}_{-1}]\\ & =& ℂ\left[e_k^{\pm 1}\right][e_1,e_2,\dots ,e_{\lfloor \frac{k}{2}\rfloor },e_{-\lfloor \frac{k-1}{2}\rfloor },\dots ,e_{-2},e_{-1}]\\ & =& ℂ\left[e_k^{\pm 1}\right][p_1,p_2,\dots ,p_{\lfloor \frac{k}{2}\rfloor },p_{-\lfloor \frac{k-1}{2}\rfloor },\dots ,p_{-2},p_{-1}]\\ & =& ℂ\left[e_k^{\pm 1}\right][p_1^{+},p_2^{+},\dots ,p_{\lfloor \frac{k}{2}\rfloor }^{+},p_{\lfloor \frac{k-1}{2}\rfloor }^{-},\dots ,p_2^{-},p_1^{-}]\text{.}\end{array}$$ For $\nu \in {\mathbb{Z}}^k$ with ${\nu }_1\ge \cdots \ge {\nu }_{\ell }>0$ define $$p_{\nu }^{+}=p_{{\nu }_1}^{+}\cdots p_{{\nu }_{\ell }}^{+}\text{and}{p}_{\nu }^{-}=p_{{\nu }_1}^{-}\cdots p_{{\nu }_{\ell }}^{-}\text{.}$$ Then $$\begin{array}{cc}C{[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]}^{S_k}\text{has basis}\left\{e_k^{\ell }{p}_{\lambda }^{+}{p}_{\mu }^{-}\hspace{0.17em}\right|\hspace{0.17em}\ell \in \mathbb{Z},\ell \left(\lambda \right)\le \lfloor \frac{k}{2}\rfloor ,\ell \left(\mu \right)\le \lfloor \frac{k-1}{2}\rfloor \}\text{.}& \text{(4.31)}\end{array}$$ In analogy with (4.12) we expect that if $R_k$ is as in Theorem 4.5 then $$\begin{array}{cc}{R}_k=C\left[e_k^{\pm 1}\right][p_1^{-},p_2^{-},\dots ]\text{.}& \text{(4.32)}\end{array}$$

The left ideal of $W_2$ generated by $E_1$ is $C\left[Y_1^{\pm 1}\right]E_1\text{.}$ This is an infinite dimensional (generically irreducible) $W_2\text{-module}$ on which $Z\left(W_2\right)$ acts by constants. It follows that $W_2$ is not a finitely generated $Z\left(W_2\right)\text{-module.}$

Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: The center by Zajj Daugherty, Arun Ram and Rahbar Virk.

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