Cohomology of complete unordered flag manifolds

Guerra, Lorenzo; Jana, Santanil

doi:10.1090/tran/9358

Cohomology of complete unordered flag manifolds
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by Lorenzo Guerra and Santanil Jana;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9358

Published electronically: January 30, 2025

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Abstract:

We consider quotients of complete flag manifolds in $\mathbb {C}^n$ and $\mathbb {R}^n$ by an action of the symmetric group on $n$ objects. We compute their cohomology with field coefficients of any characteristic. Specifically, we show that these topological spaces exhibit homological stability and we provide a closed-form description of their stable cohomology rings. We also describe a simple algorithmic procedure to determine their unstable cohomology additively.

References

A. Adem and J. Gómez, A classifying space for commutativity in Lie groups, Algebr. Geom. Topol. 15 (2015), no. 1, 493–535., DOI 10.2140/agt.2015.15.493
Michael Atiyah, The geometry of classical particles, Surveys in differential geometry, Surv. Differ. Geom., vol. 7, Int. Press, Somerville, MA, 2000, pp. 1–15. MR 1919420, DOI 10.4310/SDG.2002.v7.n1.a1
Edgar H. Brown Jr., The cohomology of $B\textrm {SO}_{n}$ and $B\textrm {O}_{n}$ with integer coefficients, Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288. MR 652459, DOI 10.1090/S0002-9939-1982-0652459-1
Thomas Church and Benson Farb, Representation theory and homological stability, Adv. Math. 245 (2013), 250–314. MR 3084430, DOI 10.1016/j.aim.2013.06.016
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 436146, DOI 10.1007/BFb0080464
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
R. H. Fox, On the Lusternik–Schnirelmann category, Ann. of Math. (2) 42 (1941), no. 2, 333–370., DOI 10.2307/1968905
Chad Giusti, Paolo Salvatore, and Dev Sinha, The mod-2 cohomology rings of symmetric groups, J. Topol. 5 (2012), no. 1, 169–198. MR 2897052, DOI 10.1112/jtopol/jtr031
Chad Giusti and Dev Sinha, Mod-two cohomology rings of alternating groups, J. Reine Angew. Math. 772 (2021), 1–51. MR 4227589, DOI 10.1515/crelle-2020-0016
Lorenzo Guerra, Hopf ring structure on the $\textrm {mod}\,p$ cohomology of symmetric groups, Algebr. Geom. Topol. 17 (2017), no. 2, 957–982. MR 3623678, DOI 10.2140/agt.2017.17.957
Lorenzo Guerra, The $\rm mod\, 2$ cohomology of the infinite families of Coxeter groups of type $B$ and $D$ as almost-Hopf rings, Algebr. Geom. Topol. 23 (2023), no. 7, 3221–3292. MR 4647676, DOI 10.2140/agt.2023.23.3221
L. Guerra and S. Jana, The mod 2 cohomology rings of the alternating subgroups of the Coxeter groups of type B, arXiv:2312.12346 (2023).
L. Guerra, S. Jana, and A. Maiti, The mod-2 cohomology groups of low-dimensional unordered flag manifolds and Auerbach bases, arXiv:2304.12990, 2024.
Lorenzo Guerra, Paolo Salvatore, and Dev Sinha, Cohomology rings of extended powers and of free infinite loop spaces, Trans. Amer. Math. Soc. 377 (2024), no. 12, 8515–8561. MR 4815515, DOI 10.1090/tran/9198
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Stanley O. Kochman, Homology of the classical groups over the Dyer-Lashof algebra, Trans. Amer. Math. Soc. 185 (1973), 83–136. MR 331386, DOI 10.1090/S0002-9947-1973-0331386-2
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974. MR 440554, DOI 10.1515/9781400881826
Rohit Nagpal, FI-modules and the cohomology of modular representations of symmetric groups, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 3358218
A. Weber and M. Wojciechowski, On the Pełczyński conjecture on Auerbach bases, Commun. Contemp. Math. 19 (2017), no. 6, 1750016., DOI 10.1142/S021919971750016X
Herbert S. Wilf, generatingfunctionology, 3rd ed., A K Peters, Ltd., Wellesley, MA, 2006. MR 2172781