Decompositions of small tensor powers and Larsen’s conjecture

Guralnick, Robert; Tiep, Pham

doi:10.1090/S1088-4165-05-00192-5

Decompositions of small tensor powers and Larsen's conjecture

Authors: Robert M. Guralnick and Pham Huu Tiep
Journal: Represent. Theory 9 (2005), 138-208
MSC (2000): Primary 20C15, 20C20, 20C33, 20C34, 20G05, 20G40
DOI: https://doi.org/10.1090/S1088-4165-05-00192-5
Published electronically: February 2, 2005
MathSciNet review: 2123127
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We classify all pairs with a closed subgroup in a classical group $\mathcal G$ with natural module over $\mathbb C$ , such that $\mathcal G$ and 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 there are no such aside from those containing the derived subgroup of $\mathcal G$ . We also find all the examples where this fails for $\dim(V)\le 6$ . 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.

[A] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, https://doi.org/10.1007/BF01388470
[AGM] M. Aschbacher, R. M. Guralnick, and K. Magaard, (in preparation).
[BL] Y. Barnea and M. Larsen, Random generation in semisimple algebraic groups over local fields, J. Algebra 271 (2004), no. 1, 1–10. MR 2022476, https://doi.org/10.1016/j.jalgebra.2002.12.001
[Atlas] 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. MR 827219
[C] 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. MR 794307
[D] Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
[Di] John D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205. MR 251758, https://doi.org/10.1007/BF01110210
[Dyn] E. B. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Soc. Translations 6 (1960), 245-378.
[FH] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
[G] 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. MR 1670777
[GLSS] 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. MR 1707675, https://doi.org/10.1006/jabr.1999.7869
[GLS] R. M. Guralnick, F. Lübeck and A. Shalev, 0,1 generation laws for Chevalley groups, (preprint).
[GMST] 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. MR 1947325, https://doi.org/10.1016/S0021-8693(02)00527-6
[GS] Robert M. Guralnick and Jan Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), no. 2, 519–571. MR 2009321, https://doi.org/10.1016/S0021-8693(03)00182-0
[GT1] R. M. Guralnick and Pham Huu Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), 4969-5023. MR
[GT2] R. M. Guralnick and Pham Huu Tiep, The non-coprime problem, (submitted).
[He] Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. Algebra 93 (1985), no. 1, 151–164. MR 780488, https://doi.org/10.1016/0021-8693(85)90179-6
[H] G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, RWTH Aachen, Germany, 1993.
[HM] 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. MR 1942256, https://doi.org/10.1112/S1461157000000711
[Him] F. Himstedt, Notes on the 2-modular irreducible representations of Steinberg's triality groups $\hspace{0.5mm}^{3}\hspace*{-0.2mm}D_{4}(q)$ , odd, (preprint).
[Ho] 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. MR 1889347
[Hu] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
[J] G. D. James, On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 417–424. MR 720791, https://doi.org/10.1017/S0305004100000803
[Jans] C. Jansen, Minimal degrees of faithful representations for sporadic simple groups and their covering groups, (in preparation).
[JLPW] 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. MR 1367961
[Jan] Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
[Ka1] 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. MR 2058618
[Ka2] N. Katz, Moments, Monodromy, and Perversity: a Diophantine Perspective, Annals of Math. Study, Princeton Univ. Press (to appear).
[KaL] William M. Kantor and Alexander Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), no. 1, 67–87. MR 1065213, https://doi.org/10.1007/BF00181465
[KL] 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. MR 1057341
[Kle] P. B. Kleidman, The Subgroup Structure of Some Finite Simple Groups, Ph.D. Thesis, Trinity College, 1986.
[KT] 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. MR 2031049, https://doi.org/10.1090/S0002-9947-03-03364-6
[LS] Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418–443. MR 360852, https://doi.org/10.1016/0021-8693(74)90150-1
[Lars] M. Larsen, A characterization of classical groups by invariant theory, (preprint, mid-1990's).
[LP] M. Larsen and R. Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. (to appear).
[LST] 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. MR 1830550, https://doi.org/10.1006/jabr.2000.8713
[Li] Martin W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), no. 3, 477–516. MR 879395, https://doi.org/10.1112/plms/s3-54.3.477
[LiS] 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. MR 2063402, https://doi.org/10.1515/jgth.2004.012
[LSh] Martin W. Liebeck and Aner Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), no. 1, 103–113. MR 1338320, https://doi.org/10.1007/BF01263616
[Lu1] Frank Lübeck, Smallest degrees of representations of exceptional groups of Lie type, Comm. Algebra 29 (2001), no. 5, 2147–2169. MR 1837968, https://doi.org/10.1081/AGB-100002175
[Lu2] Frank Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. MR 1901354, https://doi.org/10.1112/S1461157000000838
[Mag] K. Magaard, On the irreducibility of alternating powers and symmetric squares, Arch. Math. (Basel) 63 (1994), no. 3, 211–215. MR 1287249, https://doi.org/10.1007/BF01189822
[MM] K. Magaard and G. Malle, Irreducibility of alternating and symmetric squares, Manuscripta Math. 95 (1998), 169-180. MR 603309 (99a:20011)
[MMT] 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. MR 1888172, https://doi.org/10.2140/pjm.2002.202.379
[MT1] 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. MR 1889349
[MT2] K. Magaard and Pham Huu Tiep, The classes $\mathcal{C}_{6}$ and $\mathcal{C}_{7}$ of maximal subgroups of finite classical groups, (in preparation).
[Mal] Gunter Malle, Almost irreducible tensor squares, Comm. Algebra 27 (1999), no. 3, 1033–1051. MR 1669100, https://doi.org/10.1080/00927879908826479
[NRS] 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. MR 1845897, https://doi.org/10.1023/A:1011233615437
[Pre] 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. MR 905003, https://doi.org/10.1070/SM1988v061n01ABEH003200
[Sch] M. Schönert and others, GAP-Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 3rd edn., (1993-1997).
[S1] Gary M. Seitz, Some representations of classical groups, J. London Math. Soc. (2) 10 (1975), 115–120. MR 369556, https://doi.org/10.1112/jlms/s2-10.1.115
[S2] Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. MR 888704, https://doi.org/10.1090/memo/0365
[SZ] 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. MR 1223676, https://doi.org/10.1006/jabr.1993.1132
[St] Robert Steinberg, Generators for simple groups, Canadian J. Math. 14 (1962), 277–283. MR 143801, https://doi.org/10.4153/CJM-1962-018-0
[T] Pham Huu Tiep, Dual pairs and low-dimensional representations of finite classical groups, (in preparation).
[TZ1] Pham Huu Tiep and Alexander E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), no. 6, 2093–2167. MR 1386030, https://doi.org/10.1080/00927879608825690
[TZ2] 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. MR 1449955, https://doi.org/10.1006/jabr.1996.6943
[Wa1] David B. Wales, Some projective representations of 𝑆_{𝑛}, J. Algebra 61 (1979), no. 1, 37–57. MR 554850, https://doi.org/10.1016/0021-8693(79)90304-1
[Wa2] David Wales, Quasiprimitive linear groups with quadratic elements, J. Algebra 245 (2001), no. 2, 584–606. MR 1863893, https://doi.org/10.1006/jabr.2001.8935
[We] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, 1946.
[Zs] K. Zsigmondy, Zur Theorie der Potenzreste, Monath. Math. Phys. 3 (1892), 265-284.