Abstract
We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group \(G(\mathbb {F}_q)\). In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or \(G(\mathbb {F}_q)\). Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture.
Similar content being viewed by others
References
Andersen, H.H.: The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B. Ann. of Math. (2) 112(1), 113–121 (1980)
Andersen, H.H.: p-filtrations and the Steinberg module. J. Algebra 244(2), 664–683 (2001)
Bowman, C., Doty, S.R., Martin, S.: Decomposition of tensor products of modular irreducible representations for SL3. Int. Electron. J. Algebra 9, 177–219 (2011). With an appendix by C. M. Ringel
Bowman, C., Doty, S.R., Martin, S.: Decomposition of tensor products of modular irreducible representations for SL3: the p ≥ 5 case. Int. Electron. J. Algebra 17, 105–138 (2015)
Bendel, C.P., Nakano, D.K., Parshall, B.J., Pillen, C., Scott, L.L., Stewart, D.I.: Bounding extensions for finite groups and Frobenius kernels. arXiv:1208.6333 [math.RT] (2012)
Doty, S., Henke, A.: Decomposition of tensor products of modular irreducibles for SL 2. Q. J. Math. 56(2), 189–207 (2005)
Donkin, S.: A filtration for rational modules. Math. Z. 177(1), 1–8 (1981)
Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212(1), 39–60 (1993)
Drupieski, C.M.: On projective modules for Frobenius kernels and finite Chevalley groups. Bull. Lond. Math. Soc. 45(4), 715–720 (2013)
Haboush, W.J.: A short proof of the Kempf vanishing theorem. Invent. Math. 56(2), 109–112 (1980)
Humphreys, J.E.: Ordinary and modular representations of Chevalley groups. Lecture Notes in Mathematics, vol. 528. Springer-Verlag, Berlin-New York (1976)
Humphreys, J.E.: Modular representations of finite groups of Lie type, volume 326 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2006)
Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. J. Reine Angew. Math. 317, 157–199 (1980)
Jantzen, J.C.: Representations of algebraic groups, volume 107 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition (2003)
Kildetoft, T., Nakano, D.K.: On good (p,r)-filtrations for rational G-modules. J. Algebra 423, 702–725 (2015)
Mathieu, O.: Filtrations of G-modules. Ann. Sci. École Norm. Sup. (4) 23(4), 625–644 (1990)
Riche, S., Williamson, G.: arXiv:1512.08296 [math.RT] (2015)
Sobaje, P.: Varieties of G r -summands in rational G-modules. arXiv:1605.06330 [math.RT] (2016)
Steinberg, R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1963)
Wan, J., Wang, W.: The GL n (q)-module structure of the symmetric algebra around the Steinberg module. Adv. Math. 227(4), 1562–1584 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Jon F. Carlson.
Supported in part by QGM (Centre for Quantum Geometry of Moduli Spaces) funded by the Danish National Research Foundation and in part by Knut and Alice Wallenbergs Foundation
Rights and permissions
About this article
Cite this article
Kildetoft, T. Decomposition of Tensor Products Involving a Steinberg Module. Algebr Represent Theor 20, 951–975 (2017). https://doi.org/10.1007/s10468-017-9670-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-017-9670-7