## Experimental Mathematics

- Experiment. Math.
- Volume 21, Issue 2 (2012), 141-170.

### Computational Approaches to Poisson Traces Associated to Finite Subgroups of ${\rm Sp}_2(\mathbb{C})$

Pavel Etingof, Sherry Gong, Aldo Pacchiano, Qingchun Ren, and Travis Schedler

#### Abstract

We reduce the computation of Poisson traces on quotients of symplectic vector spaces by finite subgroups of symplectic automorphisms to a finite one by proving several results that bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions that are invariant under invariant Hamiltonian flow.We implement these approaches by computer together with direct computation for infinite families of groups, focusing on complex reflection and abelian subgroups of ${\rm GL}_2(\mathbb{C}) \lt {\rm Sp}_4(\mathbb{C})$, Coxeter groups of rank $\le 3$ and types $A_4$, $B_4 = C_4$, and $D_4$, and subgroups of ${\rm SL}_2(\mathbb{C})$.

#### Article information

**Source**

Experiment. Math., Volume 21, Issue 2 (2012), 141-170.

**Dates**

First available in Project Euclid: 31 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1338430827

**Mathematical Reviews number (MathSciNet)**

MR2931311

**Zentralblatt MATH identifier**

1246.53111

**Subjects**

Primary: 16S80: Deformations of rings [See also 13D10, 14D15] 17B63: Poisson algebras

**Keywords**

Poisson algebra Poisson variety Poisson traces Hamiltonian flow quotient singularity complex reflection groups

#### Citation

Etingof, Pavel; Gong, Sherry; Pacchiano, Aldo; Ren, Qingchun; Schedler, Travis. Computational Approaches to Poisson Traces Associated to Finite Subgroups of ${\rm Sp}_2(\mathbb{C})$. Experiment. Math. 21 (2012), no. 2, 141--170. https://projecteuclid.org/euclid.em/1338430827