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.
Similar content being viewed by others
References
Akyildiz, E.: Bruhat decomposition via Gm-action. Bull. Acad. Pol. Sci. Ser. Sci. Math. 28(11–12), 541–547 (1981)
Bialynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98(3), 480–497 (1973)
Billey, S.: Kostant polynomials and the cohomology ring of G/B. Duke Math. J. 96, 205–224 (1999)
Billey, S., Lakshmibai, V.: Singular loci of Schubert varieties. Prog. Math. 182, Birkhauser, Boston (2000)
Billey, S., Warrington, G.: Maximal singular loci of Schubert varieties in SL(n)/B. Trans. Am. Math. Soc. 355(10), 3915–3945 (2003)
Billey, S., Warrington, G.: Smoothness of Schubert varieties via patterns in root subsystems. Adv. Appl. Math. 34(3), 447–466 (2005)
Brion, M., Carrell, J.B.: The equivariant cohomology ring of regular varieties. Mich. Math. J. 52(1), 189–203 (2004)
Bjorner, A., Brenti, F.: Combinatorics of Coxeter Groups. Springer, Berlin (2003)
Cortez, A.: Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire. C. R. Acad. Sci. Paris Sr. I Math. 333(6), 561–566 (2001)
Collingwood, D., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Co., New York (1993)
De Mari, F., Procesi, C., Shayman, M.: Hessenberg varieties. Trans. Am. Math. Soc. 332, 529–534 (1992)
Fulman, J.: Descent identities, Hessenberg varieties, and the Weil conjectures. J. Comb. Theory Ser. A 87(2), 390–397 (1999)
Fulton, W.: Young Tableaux. With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts. Cambridge UP, Cambridge (1997)
Fulton, W.: Intersection Theory, 2nd edn. Springer, Berlin (1998)
Fung, F.Y.C.: On the topology of components of some Springer fibers and their relation to Kazhdan–Lusztig theory. Adv. Math. 178(2), 244–276 (2003)
Gasharov, V.: Sufficiency of Lakshmibai–Sandhya singularity conditions for Schubert varieties. Compositio Math. 126(1), 47–56 (2001)
Goresky, M., MacPherson, R.: On the spectrum of the equivariant cohomology ring. Canad. J. Math. 62(2), 262–283 (2010)
Harada, M., Tymoczko, J.: A positive Monk formula in the \(S\)-equivariant cohomology of type \(A\) Peterson varieties. Proc. London Math. Soc. 103(1), 40–72 (2011)
Harada, M., Tymoczko, J.: Poset pinball, GKM-compatible subspaces, and Hessenberg varieties, preprint (2010). arXiv:1007.2750
Humphreys, J.: Linear Algebraic Groups, Grad. Texts in Math. 21. Springer, New York (1964)
Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29, 2nd edn. Cambridge University Press, Cambridge (1990)
Insko, E.: Schubert calculus and the homology of the Peterson variety. Electron. J. Comb. 22(2), P2–26 (2015)
Insko, E., Tymoczko, J.: Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety, arXiv:1309.04842
Insko, E., Yong, A.: Patch ideals and Peterson varieties. Transform. Groups 17, 1011–1036 (2012)
Kassell, C., Lascoux, A., Reutenauer, C.: The singular locus of a Schubert variety. J. Algebra 269(1), 74–108 (2003)
Kostant, B.: Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \( \rho \). Sel. Math. (N. S.) 2, 43–91 (1996)
Kumar, S.: Kac Moody Groups, Their Flag Varieties, and Representation Theory. Birkhäuser, Boston (2002)
Lakshmibai, V., Sandhya, B.: Criterion for smoothness of Schubert varieties in \(Sl(n)/B\). Proc. Indian Acad. Sci. Math. Sci. 100(1), 45–52 (1990)
Manivel, L.: Le lieu singulier des variétés de Schubert. Int. Math. Res. Not. 16, 849–871 (2001)
Mbirika, A.: A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties, Electron. J. Comb. 17(1): Research Paper 153 (2010)
Peterson, D.: Quantum cohomology of \(G/P\), Lecture Course, M. I. T., Spring Term (1997)
Precup, M.: Affine pavings of Hessenberg varieties for semi simple groups. Sel. Math. (N. S.) 19(4), 903–922 (2013)
Rietsch, K.: Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Am. Math. Soc. 16(2), 363–392 (2003). (electronic)
Reitsch, K.: Quantum cohomology rings of Grassmannians and total positivity. Duke Math. J. 110(3), 523–553 (2001)
Robles, C.: Singular loci of cominuscule Schubert varieties. J. Pure Appl. Algebra 218(4), 745–759 (2014)
Springer, T.A.: Trigonometric sums, green functions of finite groups and representations of Weyl groups. Invent. Math. 36, 173–207 (1976)
Tymoczko, J.: Paving Hessenbergs by affines. Sel. Math. (N.S.) 13, 353–367 (2007)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-015-0093-5