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Decompositions of small tensor powers and Larsen’s conjecture. (English) Zbl 1109.20040
Authors’ summary: We classify all pairs \((G,V)\) with \(G\) a closed subgroup in a classical group \(\mathcal G\) with natural module \(V\) over \(\mathbb{C}\), such that \(\mathcal G\) and \(G\) have the same composition factors on \(V^{\otimes k}\) for a fixed \(k\in\{2,3,4\}\). In particular, we prove Larsen’s conjecture stating that for \(\dim(V)>6\) and \(k=4\) there are no such \(G\) aside from those containing the derived subgroup of \(\mathcal G\). We also find all the examples where this fails for \(\dim(V)\leq 6\). As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz’s recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.

MSC:
20G05 Representation theory for linear algebraic groups
20C15 Ordinary representations and characters
20C20 Modular representations and characters
20C33 Representations of finite groups of Lie type
20C34 Representations of sporadic groups
20G40 Linear algebraic groups over finite fields
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[1] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469 – 514. · Zbl 0537.20023
[2] M. Aschbacher, R. M. Guralnick, and K. Magaard, (in preparation).
[3] Y. Barnea and M. Larsen, Random generation in semisimple algebraic groups over local fields, J. Algebra 271 (2004), no. 1, 1 – 10. · Zbl 1049.20028
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · Zbl 0568.20001
[5] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. · Zbl 0567.20023
[6] Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137 – 252 (French). · Zbl 0456.14014
[7] John D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199 – 205. · Zbl 0176.29901
[8] E. B. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Translations 6 (1960), 245-378. · Zbl 0077.03403
[9] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. · Zbl 0744.22001
[10] Robert M. Guralnick, Some applications of subgroup structure to probabilistic generation and covers of curves, Algebraic groups and their representations (Cambridge, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 301 – 320. · Zbl 0973.20011
[11] Robert M. Guralnick, Martin W. Liebeck, Jan Saxl, and Aner Shalev, Random generation of finite simple groups, J. Algebra 219 (1999), no. 1, 345 – 355. · Zbl 0948.20052
[12] R. M. Guralnick, F. Lübeck and A. Shalev, 0,1 generation laws for Chevalley groups, (preprint).
[13] Robert M. Guralnick, Kay Magaard, Jan Saxl, and Pham Huu Tiep, Cross characteristic representations of symplectic and unitary groups, J. Algebra 257 (2002), no. 2, 291 – 347. · Zbl 1025.20002
[14] Robert M. Guralnick and Jan Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), no. 2, 519 – 571. · Zbl 1037.20016
[15] R. M. Guralnick and Pham Huu Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), 4969-5023. · Zbl 1062.20013
[16] R. M. Guralnick and Pham Huu Tiep, The non-coprime \(k(GV)\)problem, (submitted). · Zbl 1083.20006
[17] Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. Algebra 93 (1985), no. 1, 151 – 164. · Zbl 0583.20003
[18] G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, RWTH Aachen, Germany, 1993.
[19] Gerhard Hiss and Gunter Malle, Corrigenda: ”Low-dimensional representations of quasi-simple groups” [LMS J. Comput. Math. 4 (2001), 22 – 63; MR1835851 (2002b:20015)], LMS J. Comput. Math. 5 (2002), 95 – 126. · Zbl 1053.20504
[20] F. Himstedt, Notes on the 2-modular irreducible representations of Steinberg’s triality groups \(\hspace{0.5mm}^{3}\hspace*{-0.2mm}D_{4}(q)\), \(q\) odd, (preprint).
[21] Corneliu Hoffman, Projective representations for some exceptional finite groups of Lie type, Modular representation theory of finite groups (Charlottesville, VA, 1998) de Gruyter, Berlin, 2001, pp. 223 – 230. · Zbl 1004.20004
[22] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. · Zbl 0325.20039
[23] G. D. James, On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 417 – 424. · Zbl 0544.20011
[24] C. Jansen, Minimal degrees of faithful representations for sporadic simple groups and their covering groups, (in preparation). · Zbl 1089.20006
[25] Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer characters, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. · Zbl 0831.20001
[26] Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. · Zbl 0654.20039
[27] Nicholas M. Katz, Larsen’s alternative, moments, and the monodromy of Lefschetz pencils, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 521 – 560. · Zbl 1081.14015
[28] N. Katz, Moments, Monodromy, and Perversity: a Diophantine Perspective, Annals of Math. Study, Princeton Univ. Press (to appear). · Zbl 1079.14025
[29] William M. Kantor and Alexander Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), no. 1, 67 – 87. · Zbl 0718.20011
[30] Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. · Zbl 0697.20004
[31] P. B. Kleidman, The Subgroup Structure of Some Finite Simple Groups, Ph.D. Thesis, Trinity College, 1986.
[32] Alexander S. Kleshchev and Pham Huu Tiep, On restrictions of modular spin representations of symmetric and alternating groups, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1971 – 1999. · Zbl 1065.20013
[33] Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418 – 443. · Zbl 0325.20008
[34] M. Larsen, A characterization of classical groups by invariant theory, (preprint, mid-1990’s).
[35] M. Larsen and R. Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. (to appear). · Zbl 0778.11036
[36] Wolfgang Lempken, Bernd Schröder, and Pham Huu Tiep, Symmetric squares, spherical designs, and lattice minima, J. Algebra 240 (2001), no. 1, 185 – 208. With an appendix by Christine Bachoc and Tiep. · Zbl 1012.11054
[37] Martin W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), no. 3, 477 – 516. · Zbl 0621.20001
[38] Martin W. Liebeck and Gary M. Seitz, Subgroups of exceptional algebraic groups which are irreducible on an adjoint or minimal module, J. Group Theory 7 (2004), no. 3, 347 – 372. · Zbl 1058.20037
[39] Martin W. Liebeck and Aner Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), no. 1, 103 – 113. · Zbl 0836.20068
[40] Frank Lübeck, Smallest degrees of representations of exceptional groups of Lie type, Comm. Algebra 29 (2001), no. 5, 2147 – 2169. · Zbl 1004.20003
[41] Frank Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135 – 169. · Zbl 1053.20008
[42] K. Magaard, On the irreducibility of alternating powers and symmetric squares, Arch. Math. (Basel) 63 (1994), no. 3, 211 – 215. · Zbl 0824.20012
[43] K. Magaard and G. Malle, Irreducibility of alternating and symmetric squares, Manuscripta Math. 95 (1998), 169-180. · Zbl 0919.20009
[44] Kay Magaard, Gunter Malle, and Pham Huu Tiep, Irreducibility of tensor squares, symmetric squares and alternating squares, Pacific J. Math. 202 (2002), no. 2, 379 – 427. · Zbl 1072.20013
[45] Kay Magaard and Pham Huu Tiep, Irreducible tensor products of representations of finite quasi-simple groups of Lie type, Modular representation theory of finite groups (Charlottesville, VA, 1998) de Gruyter, Berlin, 2001, pp. 239 – 262. · Zbl 0992.20009
[46] K. Magaard and Pham Huu Tiep, The classes \(\mathcal{C}_{6}\) and \(\mathcal{C}_{7}\) of maximal subgroups of finite classical groups, (in preparation). · Zbl 0992.20009
[47] Gunter Malle, Almost irreducible tensor squares, Comm. Algebra 27 (1999), no. 3, 1033 – 1051. · Zbl 0931.20009
[48] Gabriele Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, Des. Codes Cryptogr. 24 (2001), no. 1, 99 – 121. · Zbl 1002.11057
[49] A. A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Mat. Sb. (N.S.) 133(175) (1987), no. 2, 167 – 183, 271 (Russian); English transl., Math. USSR-Sb. 61 (1988), no. 1, 167 – 183. · Zbl 0669.20035
[50] M. Schönert and others, GAP-Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 3rd edn., (1993-1997).
[51] Gary M. Seitz, Some representations of classical groups, J. London Math. Soc. (2) 10 (1975), 115 – 120. · Zbl 0333.20039
[52] Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. · Zbl 0624.20022
[53] Gary M. Seitz and Alexander E. Zalesskii, On the minimal degrees of projective representations of the finite Chevalley groups. II, J. Algebra 158 (1993), no. 1, 233 – 243. · Zbl 0789.20014
[54] Robert Steinberg, Generators for simple groups, Canad. J. Math. 14 (1962), 277 – 283. · Zbl 0103.26204
[55] Pham Huu Tiep, Dual pairs and low-dimensional representations of finite classical groups, (in preparation). · Zbl 1032.20008
[56] Pham Huu Tiep and Alexander E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), no. 6, 2093 – 2167. · Zbl 0901.20031
[57] Pham Huu Tiep and Alexander E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), no. 1, 130 – 165. · Zbl 0877.20030
[58] David B. Wales, Some projective representations of \?_{\?}, J. Algebra 61 (1979), no. 1, 37 – 57. · Zbl 0433.20010
[59] David Wales, Quasiprimitive linear groups with quadratic elements, J. Algebra 245 (2001), no. 2, 584 – 606. · Zbl 1009.20057
[60] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, 1946. · JFM 65.0058.02
[61] K. Zsigmondy, Zur Theorie der Potenzreste, Monath. Math. Phys. 3 (1892), 265-284.
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