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Generators, relations and symmetries in pairs of \(3\times 3\) unimodular matrices. (English) Zbl 1119.13004

The author describes a minimal generating set and the defining relations for the algebra of invariants in the coordinate ring \({\mathbb C}[\text{SL}(3,{\mathbb C}) \times \text{SL}(3,{\mathbb C})]\) of the variety \(\text{SL}(3,{\mathbb C}) \times \text{SL}(3,{\mathbb C})\), under the natural action of \(\text{SL}(3,{\mathbb C})\) by simultaneous conjugation. Let \(\mathcal X\) be the variety whose coordinate ring is \({\mathbb C}[{\mathcal X}]={\mathbb C}[\text{SL}(3,{\mathbb C}) \times \text{SL}(3,{\mathbb C})]^{\text{SL}(3,{\mathbb C})}\). The author shows that \(\mathcal X\) is isomorphic to a degree 6 affine hyper-surface in \({\mathbb C}^9\) which generically maps 2-to-1 onto \({\mathbb C}^8\). He describes explicitly the singular locus of \(\mathcal X\) and constructs examples of non-singular representations in the branching locus. Using the bijection \(\text{ Hom}(F_2, \text{SL}(3,{\mathbb C}))\to \text{SL}(3,{\mathbb C}) \times \text{SL}(3,{\mathbb C})\), where \(F_2\) is the free group with two generators, the author exhibits \(\text{ Out}(F_2)\)-symmetries which allow for a succinct expression of the defining relations.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
16R30 Trace rings and invariant theory (associative rings and algebras)

Software:

Mathematica
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References:

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