Annals of K-Theory

Connectedness of cup products for polynomial representations of $\mathrm{GL}_n$ and applications

Antoine Touzé

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, such as the cohomological behavior of the Schur functor.

Article information

Ann. K-Theory, Volume 3, Number 2 (2018), 287-329.

Received: 9 December 2016
Revised: 4 May 2017
Accepted: 29 May 2017
First available in Project Euclid: 4 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G10: Cohomology theory
Secondary: 18G15: Ext and Tor, generalizations, Künneth formula [See also 55U25]

cup products strict polynomial functors Steinberg's tensor product theorem Schur functor


Touzé, Antoine. Connectedness of cup products for polynomial representations of $\mathrm{GL}_n$ and applications. Ann. K-Theory 3 (2018), no. 2, 287--329. doi:10.2140/akt.2018.3.287.

Export citation


  • K. Akin, D. A. Buchsbaum, and J. Weyman, “Schur functors and Schur complexes”, Adv. in Math. 44:3 (1982), 207–278.
  • J. L. Alperin, “Diagrams for modules”, J. Pure Appl. Algebra 16:2 (1980), 111–119.
  • H. H. Andersen, “Extensions of modules for algebraic groups”, Amer. J. Math. 106:2 (1984), 489–504.
  • H. H. Andersen, “$p$-filtrations and the Steinberg module”, J. Algebra 244:2 (2001), 664–683.
  • J. Axtell, “Spin polynomial functors and representations of Schur superalgebras”, Represent. Theory 17 (2013), 584–609.
  • D. J. Benson, Representations and cohomology, I: Basic representation theory of finite groups and associative algebras, 2nd ed., Cambridge Studies in Advanced Math. 30, Cambridge University Press, 1998.
  • C. Bessenrodt and A. S. Kleshchev, “On tensor products of modular representations of symmetric groups”, Bull. London Math. Soc. 32:3 (2000), 292–296.
  • L. Breen, R. Mikhailov, and A. Touzé, “Derived functors of the divided power functors”, Geom. Topol. 20:1 (2016), 257–352.
  • J. Brundan and J. Kujawa, “A new proof of the Mullineux conjecture”, J. Algebraic Combin. 18:1 (2003), 13–39.
  • M. Chałupnik, “Koszul duality and extensions of exponential functors”, Adv. Math. 218:3 (2008), 969–982.
  • M. Chałupnik, “Derived Kan extension for strict polynomial functors”, Int. Math. Res. Not. 2015:20 (2015), 10017–10040.
  • M. Clausen, “Letter place algebras and a characteristic-free approach to the representation theory of the general linear and symmetric groups, II”, Adv. in Math. 38:2 (1980), 152–177.
  • C. W. Curtis and I. Reiner, Methods of representation theory, with applications to finite groups and orders, vol. 1, John Wiley & Sons, New York, 1981.
  • A. Djament, “Sur l'homologie des groupes unitaires à coefficients polynomiaux”, J. K-Theory 10:1 (2012), 87–139.
  • A. Djament and C. Vespa, “Sur l'homologie des groupes orthogonaux et symplectiques à coefficients tordus”, Ann. Sci. Éc. Norm. Supér. $(4)$ 43:3 (2010), 395–459.
  • S. Donkin, “On ${\rm Ext}\sp{1}$ for semisimple groups and infinitesimal subgroups”, Math. Proc. Cambridge Philos. Soc. 92:2 (1982), 231–238.
  • S. R. Doty, K. Erdmann, and D. K. Nakano, “Extensions of modules over Schur algebras, symmetric groups and Hecke algebras”, Algebr. Represent. Theory 7:1 (2004), 67–100.
  • C. M. Drupieski, “Cohomological finite-generation for finite supergroup schemes”, Adv. Math. 288 (2016), 1360–1432.
  • B. Ford and A. S. Kleshchev, “A proof of the Mullineux conjecture”, Math. Z. 226:2 (1997), 267–308.
  • V. Franjou, E. M. Friedlander, A. Scorichenko, and A. Suslin, “General linear and functor cohomology over finite fields”, Ann. of Math. $(2)$ 150:2 (1999), 663–728.
  • E. M. Friedlander, “Lectures on the cohomology of finite group schemes”, pp. 27–53 in Rational representations, the Steenrod algebra and functor homology, Panoramas et Synthèses 16, Société Mathématique de France, Paris, 2003.
  • E. M. Friedlander and A. Suslin, “Cohomology of finite group schemes over a field”, Invent. Math. 127:2 (1997), 209–270.
  • J. A. Green, Polynomial representations of ${\rm GL}_{n}$, 2nd ed., Lecture Notes in Math. 830, Springer, 2007.
  • J. Hong and O. Yacobi, “Quantum polynomial functors”, J. Algebra 479 (2017), 326–367.
  • G. D. James, “The decomposition of tensors over fields of prime characteristic”, Math. Z. 172:2 (1980), 161–178.
  • J. C. Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs 107, American Mathematical Society, Providence, RI, 2003.
  • A. S. Kleshchev and D. K. Nakano, “On comparing the cohomology of general linear and symmetric groups”, Pacific J. Math. 201:2 (2001), 339–355.
  • A. S. Kleshchev and J. Sheth, “On extensions of simple modules over symmetric and algebraic groups”, J. Algebra 221:2 (1999), 705–722.
  • H. Krause, “Koszul, Ringel and Serre duality for strict polynomial functors”, Compos. Math. 149:6 (2013), 996–1018.
  • H. Krause, “Highest weight categories and strict polynomial functors”, pp. 331–373 in Representation theory–-current trends and perspectives, edited by H. Krause et al., European Mathematical Society, Zürich, 2017.
  • N. J. Kuhn, “A stratification of generic representation theory and generalized Schur algebras”, $K$-Theory 26:1 (2002), 15–49.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, 1995.
  • S. Martin, Schur algebras and representation theory, Cambridge Tracts in Math. 112, Cambridge University Press, 1993.
  • R. Reischuk, “The adjoints of the Schur functor”, preprint, 2016.
  • A. Suslin, E. M. Friedlander, and C. P. Bendel, “Infinitesimal $1$-parameter subgroups and cohomology”, J. Amer. Math. Soc. 10:3 (1997), 693–728.
  • B. Totaro, “Projective resolutions of representations of ${\rm GL}(n)$”, J. Reine Angew. Math. 482 (1997), 1–13.
  • A. Touzé, “Cohomology of classical algebraic groups from the functorial viewpoint”, Adv. Math. 225:1 (2010), 33–68.
  • A. Touzé, “Troesch complexes and extensions of strict polynomial functors”, Ann. Sci. Éc. Norm. Supér. $(4)$ 45:1 (2012), 53–99.
  • A. Touzé, “A construction of the universal classes for algebraic groups with the twisting spectral sequence”, Transform. Groups 18:2 (2013), 539–556.
  • A. Touzé, “Ringel duality and derivatives of non-additive functors”, J. Pure Appl. Algebra 217:9 (2013), 1642–1673.
  • A. Touzé, “Bar complexes and extensions of classical exponential functors”, Ann. Inst. Fourier $($Grenoble$)$ 64:6 (2014), 2563–2637.
  • A. Touzé, “Computations and applications of some homological constants for polynomial representations of ${\rm GL}_n$”, 2017, To appear in Representations of algebras (Syracuse, NY, 2016), edited by G. Leuschke et al., Contemporary Math. 705, American Mathematical Society, Providence, RI.
  • A. Touzé, “A functorial control of integral torsion in homology”, Fund. Math. 237:2 (2017), 135–163.