Anti-commuting varieties

Chen, Xinhong; Wang, Weiqiang

doi:10.1090/tran/8017

Anti-commuting varieties
HTML articles powered by AMS MathViewer

by Xinhong Chen and Weiqiang Wang PDF

Trans. Amer. Math. Soc. 373 (2020), 1597-1617 Request permission

Abstract:

We study the anti-commuting variety which consists of pairs of anti-commuting $n\times n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT (geometric invariant theory) quotient of the anti-commuting variety with respect to the conjugation action of $GL_n$ is shown to be of pure dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe its irreducible components.

References

M. Artin, On Azumaya algebras and finite dimensional representations of rings, J. Algebra 11 (1969), 532–563. MR 242890, DOI 10.1016/0021-8693(69)90091-X
V. Baranovsky, The variety of pairs of commuting nilpotent matrices is irreducible, Transform. Groups 6 (2001), no. 1, 3–8. MR 1825165, DOI 10.1007/BF01236059
Tristan Bozec, Quivers with loops and generalized crystals, Compos. Math. 152 (2016), no. 10, 1999–2040. MR 3569998, DOI 10.1112/S0010437X1600751X
M. Brion, Introduction to action of algebraic groups, Les cours du CIRM 1(2010), 1-22.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy, Algorithms in real algebraic geometry, 2nd ed., Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006. MR 2248869
X. Chen and M. Lu, Varieties of modules over the quantum plane, preprint, 2019.
Murray Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. MR 132079, DOI 10.2307/1970336
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Ta Khongsap and Weiqiang Wang, Hecke-Clifford algebras and spin Hecke algebras. IV. Odd double affine type, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 012, 27. MR 2481480, DOI 10.3842/SIGMA.2009.012
Yiqiang Li, Canonical bases of Cartan-Borcherds type, II: Constructible functions on singular supports, Lie algebras, Lie superalgebras, vertex algebras and related topics, Proc. Sympos. Pure Math., vol. 92, Amer. Math. Soc., Providence, RI, 2016, pp. 167–179. MR 3644231
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
T. S. Motzkin and Olga Taussky, Pairs of matrices with property $L$. II, Trans. Amer. Math. Soc. 80 (1955), 387–401. MR 86781, DOI 10.1090/S0002-9947-1955-0086781-5
Alexander Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (2003), no. 3, 653–683. MR 2018787, DOI 10.1007/s00222-003-0315-6
R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327. MR 535074
Alexander H. W. Schmitt, Geometric invariant theory relative to a base curve, Algebraic cycles, sheaves, shtukas, and moduli, Trends Math., Birkhäuser, Basel, 2008, pp. 131–183. MR 2402697, DOI 10.1007/978-3-7643-8537-8_{8}
Jinkui Wan and Weiqiang Wang, Stability of the centers of group algebras of $GL_n(q)$, Adv. Math. 349 (2019), 749–780. MR 3942713, DOI 10.1016/j.aim.2019.04.026