Almost cyclic elements in Weil representations of finite classical groups

References

• Aschbacher, M. (1986). Finite Group Theory. Cambridge: Cambridge University Press.
   Google Scholar
• Blau, H. I. (1988). On linear groups with a cyclic or TI-Sylow p-subgroup. J. Algebra 114:268–285.
   Web of Science ®Google Scholar
• Bonnafé, C. (2011). The Representations of SL2(𝔽q). Heidelberg: Springer-Verlag.
   Google Scholar
• Burkhardt, R. (1976). Die Zerlegungsmatrizen der Gruppen PSL(2, pf). J. Algebra 40:75–96.
   Web of Science ®Google Scholar
• Conway, J., Curtis, R., Norton, S., Parker, R., Wilson, R. (1985). Atlas of Finite Groups. Oxford: Clarendon Press.
   Google Scholar
• Craven, D., Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p. J. Group Theory (to appear).
   Google Scholar
• Dempwolff, U. (1987). Linear groups with large cyclic subgroups and translation planes. Rend. Sem. Mat. Univ. Padova 77:69–113.
   Google Scholar
• Di Martino, L., Pellegrini, M. A., Zalesski, A. E. (2014). On generators and representations of the sporadic simple groups. Commun. Algebra 42(2):880–908.
   Web of Science ®Google Scholar
• Di Martino, L., Zalesski, A. E. (2001). Minimum polynomials and lower bounds for eigenvalue multiplicities in representations of classical groups. J. Algebra 243:228–263. Corrigendum: J. Algebra 296 (2006), 249–252.
   Web of Science ®Google Scholar
• Di Martino, L., Zalesski, A. E. (2008). Eigenvalues of unipotent elements in cross-characteristic representations of finite classical groups. J. Algebra 319:2668–2722.
   Web of Science ®Google Scholar
• Di Martino, L., Zalesski, A. E. (2012). Unipotent elements in representations of finite groups of Lie type. J. Algebra Appl. 11(2):1250038-1–1250038-25.
   Google Scholar
• Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin: De Gruyter Publ.
   Google Scholar
• Dummigan, N., Tiep, P. H. (1999). Lower bounds for minima of certain symplectic and unitary group lattices. Am. J. Math. 121:889–918.
   Web of Science ®Google Scholar
• Emmett, L., Zalesski, A. E. (2011). On regular orbits of elements of classical groups in their permutation representations. Commun. Algebra 39(9):3356–3409.
   Web of Science ®Google Scholar
• Ennola, V. (1963). On the characters of finite unitary groups. Ann. Acad. Sci. Fenn. Ser. A, I 323:120–155.
   Google Scholar
• Feit, W. (1982). The Representation Theory of Finite Groups. Amsterdam: North-Holland.
   Google Scholar
• Gérardin, P. (1977). Weil representations associated to finite fields. J. Algebra 46:54–101.
   Web of Science ®Google Scholar
• Gow, R. (2000). Commutators in finite simple groups of Lie type. Bull. London Math. Soc. 32:311–315.
   Web of Science ®Google Scholar
• Guralnick, R., Kantor, W. (2000). Probabilistic generation of finite simple groups. J. Algebra 234:743–792.
   Web of Science ®Google Scholar
• Guralnick, R., Penttila, T., Praeger, C., Saxl, J. (1999). Linear groups having certain large prime divisors. Proc. London Math. Soc. 78(3):167–214.
   Web of Science ®Google Scholar
• Guralnick, R., Saxl, J. (2003). Generation of finite almost simple groups by conjugates. J. Algebra 268:519–571.
   Web of Science ®Google Scholar
• Hering, Ch. (1974). Transitive linear groups and linear groups which contain irreducible subgroups of prime order. I. Geom. Dedicata 2:425–460.
   Google Scholar
• Hering, Ch. (1985). Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II. J. Algebra 93(1):151–164.
   Web of Science ®Google Scholar
• Hiss, G., Malle, G. (2001). Low-dimensional representations of special unitary groups. J. Algebra 236:745–767.
   Web of Science ®Google Scholar
• Hiss, G., Zalesski, A. (2009). The Weil-Steinberg character of finite classical groups with an appendix by Olivier Brunat. Represent. Theory 13:427–459.
   Google Scholar
• Huffman, W. C., Wales, D. B. (1976). Linear groups containing an element with an eigenspace of codimension two. In: Scott, R. W., Fletcher Gross, F., eds., Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975). New York: Academic Press, pp. 425–429.
   Google Scholar
• Huppert, B. (1998). Character Theory of Finite Groups. Berlin: De Gruyter Publ.
   Google Scholar
• Huppert, B. (1970). Singer-Zyklen in klassischen Gruppen. Math. Z. 117:141–150.
   Web of Science ®Google Scholar
• Huppert, B., Blackburn, N. (1982). Finite Groups II. Berlin, etc.: Springer-Verlag.
   Google Scholar
• Jansen, C., Lux, K., Parker, R., Wilson, R. (1995). An Atlas of Brauer Characters. Oxford: Oxford Science Publications, Clarendon Press.
   Google Scholar
• Kleidman, P., Liebeck, M. (1990). The Subgroup Structure of the Finite Classical Groups. Cambridge: Cambridge Univ. Press (London Math. Soc. Lecture notes no. 129).
   Google Scholar
• Landazuri, V., Seitz, G. (1999). On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32:418–443.
   Web of Science ®Google Scholar
• Lübeck, F., Malle, G. (1999). (2,3)-Generation of exceptional groups. J. London Math. Soc. 59:109–122.
   Web of Science ®Google Scholar
• Malle, G. (1990). Hurwitz groups and G2(q). Canad. Math. Bull. 33(3):349–356.
   Web of Science ®Google Scholar
• Malle, G. (1995). Small rank exceptional Hurwitz groups. In: William, M., Kantor, M. W., Martino, D. L., eds., Groups of Lie Type and Their Geometries, London Math. Soc. Lecture Notes. Vol. 207 Cambridge: Cambridge Univ. Press, pp. 173–183.
   Google Scholar
• Malle, G., Saxl, J. Weigel (1994). Generation of classical groups. Geom. Dedicata 49:85–116.
   Web of Science ®Google Scholar
• Pollatsek, H. (1976). Irreducible groups generated by transvections over finite fields of characteristic two. J. Algebra 39:328–333.
   Web of Science ®Google Scholar
• Rudloff, Ch., Zalesski, A. E. (2007). Regular submodules for cyclic Sylow p-subgroups in complex representations of classical groups. J. Group Theory 10:585–612.
   Google Scholar
• Seitz, G., Zalesskiĭ, A. E. (1993). On the minimal degrees of projective representations of the finite Chevalley groups, II. J. Algebra 158:233–243.
   Web of Science ®Google Scholar
• Serre, J.-P. (1977). Linear Representations of Finite Groups. Heidelberg: Springer-Verlag.
   Google Scholar
• Stein, A. (1998). 112-Generation of finite simple groups. Beiträge Algebra Geom. 39:349–358.
   Google Scholar
• Suprunenko, I. D. (2013). Unipotent elements of non-prime order in representations of the classical algebraic groups: Two big Jordan blocks. Zapiski Nauch. Semin. POMI 414:193–241.
   Google Scholar
• Suprunenko, I. D., Zalesski, A. E. (1998). Irreducible representations of finite classical groups containing matrices with simple spectra. Commun. Algebra 26:863–888.
   Web of Science ®Google Scholar
• Suprunenko, I. D., Zalesski, A. E. (2000). Irreducible representations of finite groups of exceptional Lie type containing matrices with simple spectra. Commun. Algebra 28(4):1789–1833.
   Web of Science ®Google Scholar
• Suzuki, M. (1982). Group Theory I. Berlin: Springer-Verlag.
   Google Scholar
• Testerman D, ., Zalesski, A. (2018). Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block. J. Group Theory 21:1–20.
   Web of Science ®Google Scholar
• Tiep, P. H., Zalesskiĭ, A. E. (1997). Some characterizations of the Weil representations of the symplectic and unitary groups. J. Algebra 192:130–165.
   Web of Science ®Google Scholar
• Tiep, P. H., Zalesskiĭ, A. E. (2000). Some aspects of finite linear groups: A survey. J. Math. Sci. 100:1893–1914. (Russian edition: Contemporary Mathematics and Its Applications. Thematic Surveys. Vol. 58, Algebra – 12.)
   Google Scholar
• Tiep, P. H., Zalesski, A. E. (2008). Hall-Higman type theorems for semisimple elements of finite classical groups. Proc. London Math. Soc. 97(3):623–668.
   Web of Science ®Google Scholar
• Wagner, A. (1980). Determination of the finite primitive reflection groups over an arbitrary field of characteristic not two, Parts I, II, III. Geom. Dedicata 9:239–253, 10 (1981) 191–203, 475–523.
   Google Scholar
• Wagner, A. (1978). Collineation groups generated by homologies of order greater than 2. Geom. Dedicata 7:387–398.
   Google Scholar
• Ward, H. N. (1972). Representations of symplectic groups. J. Algebra 20:182–195.
   Web of Science ®Google Scholar
• Zalesskiĭ, A. E. (1985). The normalizer of the extraspecial linear group (in Russian). Vesti AN BSSR Ser. Fiz.-Mat. Navuk. 6:11–16.
   Google Scholar
• Zalesski, A. (1986). Spectra of elements of order p in representations of Chevalley groups of characteristic p (in Russian). Vesti AN BSSR Ser. Fiz.-Mat. Navuk. 6:20–25.
   Google Scholar
• Zalesskiĭ, A. E. (1993). Linear groups. In: Encyclopedia of Mathematical Sciences, Vol. 37: Algebra IV. Berlin: Springer-Verlag, pp. 97–196.
   Google Scholar
• Zalesskiĭ, A. E. (1999). Minimal polynomials and eigenvalues of p-elements in representations of groups with a cyclic Sylow p-subgroup. J. London Math. Soc. 59:845–866.
   Web of Science ®Google Scholar
• Zalesski, A. E. (2006). The number of distinct eigenvalues of elements in finite linear groups. J. London Math. Soc. Part 2 74:361–378.
   Web of Science ®Google Scholar
• Zalesski, A. E. (2008). Minimal polynomials of the elements of prime order in complex irreducible representations of quasi-simple groups. J. Algebra 320:2496–2525.
   Web of Science ®Google Scholar
• Zalesski, A. E. (2009). On eigenvalues of group elements in representations of simple algebraic groups and finite Chevalley groups. Acta Appl. Math. 108:175–195.
   Web of Science ®Google Scholar
• Zalesskiĭ, A. E., Serezhkin, V. N. (1976). Linear groups generated by transvections. Math. USSR Izvestija 10:25–46.
   Web of Science ®Google Scholar
• Zalesskiĭ, A. E., Serezhkin, V. N. (1981). Linear groups generated by reflections. Math. USSR Izvestija 17:477–503.
   Web of Science ®Google Scholar
• Zsigmondy, K. (1982). Zur Theorie der Potenzreste. Monatsh. für Math. Phys. 3:265–284.
   Google Scholar