Abstract
Precup recently proved that intersections with Schubert cells pave regular nilpotent Hessenberg varieties. We use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explicitly identify some of the coefficients of Schubert classes in this expansion, answering a problem of independent interest in Schubert calculus. We also identify some singular points in a certain family of Schubert varieties in general Lie type.
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Acknowledgments
The authors are thankful to Dave Anderson, Sam Evens, Megumi Harada, Nicholas Teff, and aBa Mbirika for many helpful conversations during this project. The suggestions of an anonymous referee also greatly improved this paper. EI was partially supported by NSF VIGRE grant DMS-0602242. JT was partially supported by NSF grants DMS-0801554 and DMS-1248171, and as an Alfred P. Sloan Research Fellow.
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Insko, E., Tymoczko, J. Intersection theory of the Peterson variety and certain singularities of Schubert varieties. Geom Dedicata 180, 95–116 (2016). https://doi.org/10.1007/s10711-015-0093-5
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DOI: https://doi.org/10.1007/s10711-015-0093-5