Classical type data

## Classical type data

Let ${W}_{0}$ be the group of $r×r$ matrices generated by the symmetric group of permutation matrices ${S}_{r}$ and the additional matrix ${s}_{r}=\mathrm{diag}\left(1,\dots ,1,-1\right).$ The group ${W}_{0}$ acts on the polynomial ring $ℂ\left[{h}_{1},\dots ,{h}_{r}\right]$ by and $w\left({p}_{1}{p}_{2}\right)=\left(w{p}_{1}\right)\left(w{p}_{2}\right)$ for $w\in {W}_{0}$ and ${p}_{1},{p}_{2}\in ℂ\left[{h}_{1},\dots ,{h}_{r}\right].$ Let and define elements ${z}_{V}^{\left(l\right)}\in C,l\in {ℤ}_{\ge 0},$ by setting $z V u = ∑ l∈ ℤ ≥0 z V l u -l ,$ and $z V u +u- 1 2 = u+ 1 2 u+ 1 2 y u+ 1 2 y-r u- 1 2 y u- 1 2 y+r ∏ i=1 r u+ 1 2 + h i u+ 1 2 - h i u- 1 2 - h i u- 1 2 + h i .$

## Notes and References

See [Bou, Ch. I §3 Prop. 11] for the fact that the Casimir element is in the center of $U𝔤$.

## Bibliography

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[Dr] V.G. Drinfeld, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342. MR1025154 (91b:16046)

[LR] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl and Type A Iwahori-Hecke algebras, Advances Math. 125 (1997), 1-94. MR1427801 (98c:20015)