Conjugate Frobenius structures on the orbit of linear presentations.
Author
Al-Maamariyah, Zainab Saleh Rashid.
Publisher
Sultan Qaboos University.
English abstract
The notion of Frobenius manifold was introduced by Boris Dubrovin as a geo metric realization of a potential F that satisfies a system of partial differential
equations known in topological fields theory as Witten-Dijkgraaf-Verlinde Verlinde (WDVV) equations. The orbit space of an irreducible linear rep resentation of a finite group is a variety whose coordinate ring is the ring
of invariant polynomials. Boris Dubrovin proved that the orbit space of the
standard reflection representation of an irreducible finite Coxeter group W
acquires a natural polynomial Frobenius manifold structure. We formulate
and apply Dubrovin's method on various orbits spaces of linear representa tions of finite groups. We find some of them have none or several natural
Frobenius manifold structures. These Frobenius manifold structures include
rational and trivial structures which can be constructed on the orbit space
of some finite groups. As a major consequence of our approach, we identify
a new conjugacy relation on a certain type of flat pencil of metric which re sults in a conjugacy relation between particular classes of Frobenius manifold
structures. It leads to a geometric interpretation for the inversion symmetry
of solutions to WDVV equations.
