Abstract
Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G-modules. A key question related to these conjectures is whether the tensor product of the rth Steinberg module with a simple module with prth restricted highest weight admits a good filtration. In this paper we verify this statement (i) when p ≥ 2h − 4 (h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.
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References
H. H. Andersen, The first cohomology group of a line bundle on G/B, Invent. Math. 51 (1979), 287–296.
H. H. Andersen, The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B, Ann. of Math 112 (1980), 113–121.
H. H. Andersen, p-filtrations and the Steinberg module, J. Algebra 244 (2001), 664–683.
H. H. Andersen, The Steinberg linkage class for a reductive algebraic group, Ark. Mat. 56 (2018), no. 2, 229–241.
H. H. Andersen, J. C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487–525.
S. Donkin, A filtration for rational modules, Math. Zeit. 177 (1981), 1–8.
S. Donkin, On tilting modules for algebraic groups, Math. Zeit. 212 (1993), 39–60.
S. Donkin, A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup, J. Algebra 519 (2019), 253–272.
S. Garibaldi, R. M. Guralnick, D. K. Nakano, Globally irreducible Weyl modules, J. Algebra 477 (2017), 69–87.
M. Gros, M. Kaneda, Contraction par Frobenius et modules de Steinberg, Ark. Mat. 56 (2018), 319–332.
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.
J. C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr. 67 (1973), v+124 pp.
J. C. Jantzen, First cohomology groups for classical Lie algebras, in: Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progress in Mathematics, Vol. 95, Birkhäuser, Basel, 1991, pp. 289–315.
J. C. Jantzen, Representations of Algebraic Groups, 2nd Ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.
T. Kildetoft, D. K. Nakano, On good (p, r) filtrations for rational G-modules, J. Algebra 423 (2015), 702–725.
M. A. A. van Leeuwen, A. M. Cohen, B. Lisser, LiE, A Package for Lie Group Computations, Computer Algebra Nederland, Amsterdam, 1992.
F. Lübeck, Tables of Weight Multiplicities, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/index.html.
W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 3/4 (1997), no. 24, 235–265.
O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole Norm. Sup. 23 (1990), 625–644.
B. J. Parshall, L. L. Scott, On p-filtrations of Weyl modules, J. London Math. Soc. 91 (2015), 127–158.
C. Pillen, Tensor product of modules with restricted highest weight, Comm. Algebra 21 (1993), 3647–3661.
A. A. Premet, I. D. Suprunenko, The Weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights, Comm. Algebra 11 (1983), 1309–1342.
S. Riche, G. Williamson, Tilting modules and the p-canonical basis, Astérisque (2018), no. 397, ix+184 pp.
C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223.
L. L. Scott, Representations in characteristic p, in: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., Vol. 37, Amer. Math. Soc., Providence, RI, 1980, pp. 319–331.
P. Sobaje, On (p, r)-filtrations and tilting modules, Proc. Amer. Math. Soc. 146 (2018), no. 5, 1951–1961.
University of Georgia VIGRE Algebra Group, First cohomology of finite groups of Lie type: simple modules with small dominant weights, Transactions of the AMS 365 (2013), 1025–1050.
G. Williamson, Schubert calculus and torsion explosion, J. Amer. Math. Soc. 30 (2017), no. 4, 1023–1046.
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BENDEL, C.P., NAKANO, D.K., PILLEN, C. et al. ON TENSORING WITH THE STEINBERG REPRESENTATION. Transformation Groups 25, 981–1008 (2020). https://doi.org/10.1007/s00031-019-09530-x
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DOI: https://doi.org/10.1007/s00031-019-09530-x