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Minimal polynomials of elements of order $p$ in $p$-modular projective representations of alternating groups


Authors: A. S. Kleshchev and A. E. Zalesski
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
Published electronically: October 21, 2003
MathSciNet review: 2051120
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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 $3\cdot S_7$.


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  • 1. D. Benson, Spin modules for symmetric groups, J. London Math. Soc. (2) 38 (1988), 250-262. MR 89k:20020
  • 2. B. Huppert and N. Blackburn, Finite groups II, Grundlehren der mathematischen Wissenschaften, Band 242, Springer-Verlag, Berlin, 1982. MR 84i:20001a
  • 3. A. Chermak, Quadratic Pairs, preprint, 2001.
  • 4. Ch. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, John Wiley and Sons, New York-London, 1962. MR 26:2519
  • 5. W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982. MR 83g:20001
  • 6. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, Vol. 682, Springer-Verlag, Berlin-New York, 1978. MR 80g:20019
  • 7. A. S. Kleshchev, Branching rules for modular representations of symmetric groups, II, J. reine angew. Math. 459 (1995), 163-212. MR 96m:20019b
  • 8. A. S. Kondratiev and A. E. Zalesski, Linear groups of degree at most $27$ over residue rings modulo $p^{k}$, J. Algebra 240 (2001), 120-142. MR 2002c:20076
  • 9. C. Jansen, K. Lux, R. A. Parker, and R. A. Wilson, An atlas of Brauer characters, The Clarendon Press, Oxford University Press, New York, 1995. MR 96k:20016
  • 10. S. D. Smith, Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), 286-289. MR 83g:20043
  • 11. D. Wales, Some projective representations of $S_{n}$, J. Algebra 61 (1979), 37-57. MR 81f:20015
  • 12. A. E. Zalesski{\u{\i}}\kern.15em, 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), 845-866. MR 2001a:20018
  • 13. A. E. Zalesski{\u{\i}}\kern.15em, Eigenvalues of prime-order elements in projective representations of alternating groups, Vestsi Akad. Navuk Belarusi, ser. Fiz.-Mat-Inform. (1996), no. 3, 41-43 (in Russian). MR 98b:20017

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Additional Information

A. S. Kleshchev
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: klesh@math.uoregon.edu

A. E. Zalesski
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, England
Email: a.zalesskii@uea.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-03-07242-3
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
Article copyright: © Copyright 2003 American Mathematical Society

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