Minimal polynomials of elements of order $p$ in $p$-modular projective representations of alternating groups
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- by A. S. Kleshchev and A. E. Zalesski PDF
- Proc. Amer. Math. Soc. 132 (2004), 1605-1612 Request permission
Abstract:
Let $F$ be an algebraically closed field of characteristic $p>0$ and let $G$ be a quasi-simple group with $G/Z(G)\cong A_n$. We describe the minimal polynomials of elements of order $p$ in irreducible representations of $G$ over $F$. If $p=2$, we determine the minimal polynomials of elements of order $4$ in $2$-modular irreducible representations of $A_{n}$, $S_n$, $3\cdot A_6$, $3\cdot S_6$, $3\cdot A_7$, and $3\cdot S_7$.References
- Dave Benson, Spin modules for symmetric groups, J. London Math. Soc. (2) 38 (1988), no. 2, 250–262. MR 966297, DOI 10.1112/jlms/s2-38.2.250
- Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
- A. Chermak, Quadratic Pairs, preprint, 2001.
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
- G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
- A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, DOI 10.1006/jabr.1995.1362
- A. S. Kondratiev and A. E. Zalesskiĭ, Linear groups of degree at most 27 over residue rings modulo $p^k$, J. Algebra 240 (2001), no. 1, 120–142. MR 1830547, DOI 10.1006/jabr.2000.8739
- 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
- Stephen D. Smith, Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), no. 1, 286–289. MR 650422, DOI 10.1016/0021-8693(82)90076-X
- David B. Wales, Some projective representations of $S_{n}$, J. Algebra 61 (1979), no. 1, 37–57. MR 554850, DOI 10.1016/0021-8693(79)90304-1
- A. E. Zalesskiĭ, Minimal polynomials and eigenvalues of $p$-elements in representations of quasi-simple groups with a cyclic Sylow $p$-subgroup, J. London Math. Soc. (2) 59 (1999), no. 3, 845–866. MR 1709084, DOI 10.1112/S0024610799007498
- A. E. Zalesskiĭ, Eigenvalues of prime-order elements in projective representations of alternating groups, Vestsī Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk 3 (1996), 41–43, 131 (Russian, with English and Russian summaries). MR 1446938
Additional Information
- A. S. Kleshchev
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 268538
- Email: klesh@math.uoregon.edu
- A. E. Zalesski
- Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, England
- MR Author ID: 196858
- Email: a.zalesskii@uea.ac.uk
- Received by editor(s): November 18, 2002
- Received by editor(s) in revised form: February 19, 2003
- Published electronically: October 21, 2003
- Communicated by: Stephen D. Smith
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1605-1612
- MSC (2000): Primary 20C30; Secondary 20C20, 20D06
- DOI: https://doi.org/10.1090/S0002-9939-03-07242-3
- MathSciNet review: 2051120