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Frobenius Manifolds on Orbits Spaces

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Abstract

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group \({\mathcal {W}}\) acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.

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Acknowledgements

The authors thank Robert Howett, Hans-Christian Herbig and Christopher Seaton for useful discussions. They very much appreciate the Magma program’s support team for their helpful cooperation.

Funding

This work was partially funded by the internal grant of Sultan Qaboos University (Grant No. IG/SCI/DOMS/19/08).

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Sultan Qaboos University, Muscat, Oman

    Zainab Al-Maamari & Yassir Dinar

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  1. Zainab Al-Maamari
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  2. Yassir Dinar
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Correspondence to Yassir Dinar.

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The authors have no relevant financial or non-financial interests to disclose.

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Communicated by Youjin Zhang.

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Al-Maamari, Z., Dinar, Y. Frobenius Manifolds on Orbits Spaces. Math Phys Anal Geom 25, 22 (2022). https://doi.org/10.1007/s11040-022-09434-5

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