Hostname: page-component-cb9f654ff-nr592 Total loading time: 0 Render date: 2025-08-24T18:13:46.122Z Has data issue: false hasContentIssue false
  • English
  • Français

POTENTIALS OF A FROBENIUS-LIKE STRUCTURE

Part of: Basic linear algebra Symplectic geometry, contact geometry Singularities Designs and configurations

Published online by Cambridge University Press:  28 January 2018

CLAUS HERTLING  and
ALEXANDER VARCHENKO
CLAUS HERTLING
Affiliation:
Lehrstuhl für Mathematik VI, Universität Mannheim A5,6, 68131 Mannheim, Germany e-mail: hertling@math.uni-mannheim.de
ALEXANDER VARCHENKO
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3250, USA e-mail: anv@email.unc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

MSC classification

Primary: 05B35: Matroids, geometric lattices
Secondary: 15A03: Vector spaces, linear dependence, rank 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 32S22: Relations with arrangements of hyperplanes

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 3 , September 2018 , pp. 681 - 693
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Dubrovin, B., Geometry of 2D topological field theories, in Integrable systems and quantum groups, Lecture notes in mathematics, 1620 (Francaviglia, M. and Greco, S., Editors) (Springer, Berlin Heidelberg, 1996), 120–348.Google Scholar
2. Edmonds, J., Matroid partition, in Mathematics of the decision sciences: Part 1 (Dantzig, G. B. and Veinott, A.F., Editors) (AMS, 1968), 335–345. Reprinted in 50 years of integer programming 1958–2008 (Jünger, M., et al. Editors) (Springer, Berlin Heidelberg, 2010), 207–217.Google Scholar
3. Hertling, C. and Varchenko, A., Potentials of a Frobenius type structure and m bases of a vector space, arXiv:1608.08423, 18 p.Google Scholar
4. Manin, Y. I., Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society colloquium publications, vol. 47 (AMS, Providence, RI, 1999).Google Scholar
5. Prudhom, A. and Varchenko, A., Potentials of a family of arrangements of hyperplanes and elementary subarrangements, arXiv:1611.03944, 24 p.Google Scholar
6. Varchenko, A., Arrangements and Frobenius like structures, Ann. Facul. Sci. Toulouse Ser. 6, 24 (1) (2015), 133–204.Google Scholar
7. Varchenko, A., Critical set of the master function and characteristic variety of the associated Gauss–Manin differential equations, Arnold Math. J. 1 (3) (2015), 253–282, DOI 10.1007/s40598-015-0020-8.Google Scholar
8. Varchenko, A., On axioms of Frobenius like structure in the theory of arrangements, J. Integr. 1 (1) (2016), https://doi.org/10.1093/integr/xyw007Google Scholar