Equations and syzygies of some Kalman varieties

Sam, Steven

doi:10.1090/S0002-9939-2012-11593-X

Equations and syzygies of some Kalman varieties
HTML articles powered by AMS MathViewer

by Steven V Sam

Proc. Amer. Math. Soc. 140 (2012), 4153-4166

DOI: https://doi.org/10.1090/S0002-9939-2012-11593-X

Published electronically: April 26, 2012

PDF | Request permission

Abstract:

Given a subspace $L$ of a vector space $V$, the Kalman variety consists of all matrices of $V$ that have a nonzero eigenvector in $L$. Ottaviani and Sturmfels described minimal equations in the case that $\dim L = 2$ and conjectured minimal equations for $\dim L = 3$. We prove their conjecture and describe the minimal free resolution in the case that $\dim L = 2$, as well as some related results. The main tool is an exact sequence which involves the coordinate rings of these Kalman varieties and the normalizations of some related varieties. We conjecture that this exact sequence exists for all values of $\dim L$.

References

Christine Berkesch, Daniel Erman, Manoj Kummini, Steven V Sam, Poset structures in Boij–Söderberg theory, to appear in Int. Math. Res. Not. IMRN. arXiv:1010.2663v1.
Giandomenico Boffi, The universal form of the Littlewood-Richardson rule, Adv. in Math. 68 (1988), no. 1, 40–63. MR 931171, DOI 10.1016/0001-8708(88)90007-2
Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2107324
David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4397–4426. MR 1990756, DOI 10.1090/S0002-9947-03-03291-4
Daniel R. Grayson, Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, version 1.4. Available at http://www.math.uiuc.edu/Macaulay2/.
R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana (2) 5 (1960), 102–119. MR 127472
Giorgio Ottaviani, Bernd Sturmfels, Matrices with eigenvectors in a given subspace, to appear in Proc. Amer. Math. Soc. arXiv:1012.1016v1.
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690, DOI 10.1017/CBO9780511546556