The Casimir

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

Last updates: 8 October 2010

Classical type data

Let $W_0$ be the group of $r\times r$ 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 $$w{h}_i=h_{w\left(i\right)},\text{ for }w\in S_r,\text{and}{s}_r{h}_j=\left\{\begin{array}{r}{h}_j,\text{if}j\ne r,\\ -h_r,\text{if}j=r,\end{array}\right.$$ 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 $$C={ℂ\left[h_1,\dots ,h_r\right]}^{W_0}=\left\{p\in ℂ\left[h_1,\dots ,h_r\right]|wp=p\text{ for }w\in W_0\right\}$$ and define elements $z_V^{\left(l\right)}\in C,l\in {\mathbb{Z}}_{\ge 0},$ by setting $$z_V\left(u\right)=\sum _{l\in {\mathbb{Z}}_{\ge 0}}{z}_V^{\left(l\right)}{u}^{-l},$$ and $$z_V\left(u\right)+u-\frac{1}{2}=\left(u+\frac{1}{2}\right)\frac{\left(u+\frac{1}{2}y\right)\left(u+\frac{1}{2}y-r\right)}{\left(u-\frac{1}{2}y\right)\left(u-\frac{1}{2}y+r\right)}\prod _{i=1}^r\frac{\left(u+\frac{1}{2}+h_i\right)\left(u+\frac{1}{2}-h_i\right)}{\left(u-\frac{1}{2}-h_i\right)\left(u-\frac{1}{2}+h_i\right)}.$$

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)

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