## 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 $C\left[{y}_{1},\dots ,{y}_{k}\right]$ is a subalgebra of ${𝒲}_{k}$. The symmetric group ${S}_{k}$ acts on $C\left[{y}_{1},\dots ,{y}_{k}\right]$ by permuting the variables and the ring of symmetric functions is $C[y1,…, yk] Sk = { f∈ C[y1,…, yk] | w p=p, for w∈Sk }.$ A classical fact (see, for example [Kl, Theorem 3.3.1]) is that the center of the degenerate affine Hecke algebra ${ℋ}_{k}$ is $Z(ℋk) = C[y1,…, yk] Sk .$ 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 $ℛk = {f∈ C[y1,…, yk] Sk | f(y1, -y1,y3, …,yk) = f(0,0,y3, …,yk)}.$

Proof. Step 1: $f$ commutes with all ${y}_{i}$ $⇔$ $f\in C\left[{y}_{1},\dots ,{y}_{k}\right]$:
Assume $f\in {𝒲}_{k}$ and write $f= ∑ cd n1,…, nk d n1,…, nk$ 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 $\left(i,j\right)$ of $d$ such that $j\ne i\prime$. Then, by (3.5) and (3.6) (ELSEWHERE) $the coefficient of yi d n1,…, nk in yif is cd n1,…, nk$ and $the coefficient of yi d n1,…, nk in fyi is 0.$ If ${y}_{i}f=f{y}_{i}$, it follows that there is no such edge, and so $d=1$. Thus $f\in C\left[{y}_{1},\dots ,{y}_{k}\right]$. Conversely, if $f\in C\left[{y}_{1},\dots ,{y}_{k}\right]$ then ${y}_{i}f=f{y}_{i}$.

Step 2. $f\in C\left[{y}_{1},\dots ,{y}_{k}\right]$ commutes with all ${t}_{{s}_{i}}$ $⇔$ $f\in {ℛ}_{k}$ :
Assume $f\in C\left[{y}_{1},\dots ,{y}_{k}\right]$ and write

 $f=\sum _{a,b\in {ℤ}_{\ge 0}}{y}_{1}^{a}{y}_{2}^{b}{f}_{a,b}$,    where ${f}_{a,b}\in C\left[{y}_{3},\dots ,{y}_{k}\right]$.
Then $f\left(0,0,{y}_{3},\dots {y}_{k}\right)=\sum _{a,b\in {ℤ}_{\ge 0}}{f}_{a,b}$ and
 $f\left({y}_{1},-{y}_{1},{y}_{3},\dots ,{y}_{k}\right)=\sum _{a,b\in {ℤ}_{\ge 0}}{\left(-1\right)}^{b}{y}_{1}^{a+b}{f}_{a,b}=\sum _{\ell \in {ℤ}_{\ge 0}}{y}_{1}^{\ell }\left(\sum _{b=0}^{\ell }{\left(-1\right)}^{b}{f}_{\ell -b,b}\right)$. (4.9)
By direct computation using (3.11) and (3.12),
 ${t}_{{s}_{1}}{y}_{1}^{a}{y}_{2}^{b}={s}_{1}\left({y}_{1}^{a}{y}_{2}^{b}\right){t}_{{s}_{1}}-\frac{{y}_{1}^{a}{y}_{2}^{b}-{s}_{1}\left({y}_{1}^{a}{y}_{2}^{b}\right)}{{y}_{1}-{y}_{2}}+{\left(-1\right)}^{a}\sum _{r=1}^{a+b}{\left(-1\right)}^{r}{y}_{1}^{a+b-r}{e}_{1}{y}_{1}^{r-1}$,
and it follows that
 ${t}_{{s}_{1}}f=\left({s}_{1}f\right){t}_{{s}_{1}}-\frac{f-{s}_{1}f}{{y}_{1}-{y}_{2}}+\sum _{\ell \in {ℤ}_{>0}}\left(\left(\sum _{r=1}^{\ell }{\left(-1\right)}^{r}{y}_{1}^{\ell -r}{e}_{1}{y}_{1}^{r-1}\right)\left(\sum _{b=0}^{\ell }{\left(-1\right)}^{\ell -b}{f}_{\ell -b,b}\right)\right)$. (4.10)
Hence, if $f\in {C\left[{y}_{1},\dots ,{y}_{k}\right]}^{{S}_{k}}$ and $f\left({y}_{1},-{y}_{1},{y}_{3},\dots ,{y}_{k}\right)=f\left(0,0,{y}_{3},\dots ,{y}_{k}\right)$ 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 $f\in C\left[{y}_{1},\dots ,{y}_{k}\right]$ and ${t}_{{s}_{i}}f=f{t}_{{s}_{i}}$ then
 ${s}_{i}f=f$     and     $\sum _{b=0}^{\ell }{\left(-1\right)}^{\ell -b}{f}_{\ell -b,b}=0$,   for  $\ell \ne 0$,
so that $f\in {C\left[{y}_{1},\dots ,{y}_{k}\right]}^{{S}_{k}}$ and $f\left({y}_{1},-{y}_{1},{y}_{3},\dots ,{y}_{k}\right)=f\left(0,0,{y}_{3},\dots ,{y}_{k}\right)$. □

The power sum symmetric functions ${p}_{i}$, $i\in {ℤ}_{>0}$, 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 }\left(y;t\right)={P}_{\lambda }\left({y}_{1},\dots ,{y}_{k};t\right)=\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,
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 ${P}_{\lambda }\left(y;t\right)$ is equal to 1. The Schur Q-functions (see [Mac, Ch. III (8.7)]) are
 ${Q}_{\lambda }=\left\{\begin{array}{ll}0,& \text{if}\phantom{\rule{0.5em}{0ex}}\lambda \phantom{\rule{0.5em}{0ex}}is not strict,\\ {2}^{\ell \left(\lambda \right)}{P}_{\lambda }\left(y;-1\right),& \text{if}\phantom{\rule{0.5em}{0ex}}\lambda \phantom{\rule{0.5em}{0ex}}is strict,\end{array}$
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])
 ${ℛ}_{k}=C\left[{p}_{1},{p}_{3},{p}_{5},\dots \right]=C\text{-span}\left\{{Q}_{\lambda }\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\lambda \phantom{\rule{0.5em}{0ex}}\text{is strict}\right\}$. (4.12)
More generally, let $r\in {ℤ}_{>0}$ and let $\zeta$ be a primitive $r$th root of unity. Define
 ${ℛ}_{r,k}=\left\{f\in {ℤ\left[\zeta \right]\left[{y}_{1},\dots ,{y}_{k}\right]}^{{S}_{k}}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}f\left({y}_{1},\zeta {y}_{1},\dots {\zeta }^{r-1}{y}_{1},{y}_{r+1},\dots ,{y}_{k}\right)=f\left(0,0,\dots 0,{y}_{r+1},\dots ,{y}_{k}\right)\right\}$.
Then
 ${ℛ}_{r,k}{\otimes }_{ℤ\left[\zeta \right]}ℚ\left(\zeta \right)=ℚ\left(\zeta \right)\left[{p}_{i}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}i\ne 0\phantom{\rule{0.2em}{0ex}}\mathrm{mod}\phantom{\rule{0.2em}{0ex}}r\right]$, (4.13)
and ${ℛ}_{r,k}$ has $ℤ\left[\zeta \right]$-bases
 $\left\{{P}_{\lambda }\left(y;\zeta \right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}{m}_{i}\left(\lambda \right), (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

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[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??????