The Witt rings of many flag varieties are exterior algebras

Hemmert, Tobias; Zibrowius, Marcus

doi:10.1090/tran/9188

The Witt rings of many flag varieties are exterior algebras
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by Tobias Hemmert and Marcus Zibrowius;

Trans. Amer. Math. Soc. 377 (2024), 6427-6463

DOI: https://doi.org/10.1090/tran/9188

Published electronically: June 18, 2024

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Abstract:

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types $G_2$ and $F_4$. The results also extend to flag varieties over other algebraically closed fields.

References

J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 252560
Paul Balmer, Witt groups, Handbook of $K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 539–576. MR 2181829, DOI 10.1007/978-3-540-27855-9_{1}1
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
Baptiste Calmès and Viktor Petrov, WeylGroups: root systems and Weyl groups. Version 0.5.2, A Macaulay2 package, 2021, https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages.
Donald M. Davis, Representation types and 2-primary homotopy groups of certain compact Lie groups, Homology Homotopy Appl. 5 (2003), no. 1, 297–324. MR 2006403, DOI 10.4310/HHA.2003.v5.n1.a13
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
David González-Álvaro and Marcus Zibrowius, The stable converse soul question for positively curved homogeneous spaces, J. Differential Geom. 119 (2021), no. 2, 261–307. MR 4318296, DOI 10.4310/jdg/1632506394
Skip Garibaldi and Michael Carr, Geometries, the principle of duality, and algebraic groups, Expo. Math. 24 (2006), no. 3, 195–234. MR 2250947, DOI 10.1016/j.exmath.2005.11.001
Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
Tobias Hemmert, KO-theory of complex flag varieties, Ph.D. Thesis, Heinrich Heine University Düsseldorf, 2018.
Tobias Hemmert, KO-theory of complex flag varieties of ordinary type, arXiv:1902.00713, 2019
Thomas Hudson, Arthur Martirosian, and Heng Xie, Witt groups of spinor varieties, Proc. Lond. Math. Soc. (3) 125 (2022), no. 5, 1152–1178. MR 4508341, DOI 10.1112/plms.12479
Luke Hodgkin, The equivariant Künneth theorem in $K$-theory, Topics in $K$-theory. Two independent contributions, Lecture Notes in Math., Vol. 496, Springer, Berlin-New York, 1975, pp. 1–101. MR 478156
Tobias Hemmert and Marcus Zibrowius, Fixed cones in weight lattices, Macualay2-program, 2023, \PrintDOI{10.5281/zenodo.10062260}.
Richard Kane, Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 5, Springer-Verlag, New York, 2001. MR 1838580, DOI 10.1007/978-1-4757-3542-0
Akira Kono and Shin-ichiro Hara, $K\textrm {O}$-theory of Hermitian symmetric spaces, Hokkaido Math. J. 21 (1992), no. 1, 103–116. MR 1153756, DOI 10.14492/hokmj/1381413257
Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. MR 114875, DOI 10.2307/2372999
Kamran Alizadeh Rad, KO-theorie komplexer Fahnenvarietäten vom Dynkin-typ E, Master’s Thesis, Heinrich Heine University Düsseldorf, 2019.
Jean-Pierre Serre, Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts, Sém. Bourbaki 2 (1951–1954), 447–454
Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, RI, 1968. MR 230728
David A. Vogan Jr., Three-dimensional subgroups and unitary representations, Challenges for the 21st century (Singapore, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 213–250. MR 1875021
Heng Xie, An application of Hermitian $K$-theory: sums-of-squares formulas, Doc. Math. 19 (2014), 195–208. MR 3178250, DOI 10.4171/dm/444
Marcus Zibrowius, Witt groups of complex cellular varieties, Doc. Math. 16 (2011), 465–511. MR 2823367, DOI 10.4171/dm/339
Marcus Zibrowius, Twisted Witt groups of flag varieties, J. K-Theory 14 (2014), no. 1, 139–184. MR 3238260, DOI 10.1017/is014003028jkt260
Marcus Zibrowius, KO-rings of full flag varieties, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2997–3016. MR 3301890, DOI 10.1090/S0002-9947-2014-06318-6
Marcus Zibrowius, Dual involutions in finite Coxeter groups, Comm. Algebra 52 (2024), no. 3, 1028–1037. MR 4709577, DOI 10.1080/00927872.2023.2255663

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The Witt rings of many flag varieties are exterior algebras
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Abstract: