On an asymptotic behavior of elements of order 𝑝 in irreducible representations of the classical algebraic groups with large enough highest weights

Suprunenko, I.

doi:10.1090/S0002-9939-01-05934-2

On an asymptotic behavior of elements of order $p$ in irreducible representations of the classical algebraic groups with large enough highest weights
HTML articles powered by AMS MathViewer

by I. D. Suprunenko PDF

Proc. Amer. Math. Soc. 129 (2001), 2581-2589 Request permission

Abstract:

The behavior of the images of a fixed element of order $p$ in irreducible representations of a classical algebraic group in characteristic $p$ with highest weights large enough with respect to $p$ and this element is investigated. More precisely, let $G$ be a classical algebraic group of rank $r$ over an algebraically closed field $K$ of characteristic $p>2$. Assume that an element $x\in G$ of order $p$ is conjugate to that of an algebraic group of the same type and rank $m<r$ naturally embedded into $G$. Next, an integer function $\sigma _x$ on the set of dominant weights of $G$ and a constant $c_x$ that depend only upon $x$, and a polynomial $d$ of degree one are defined. It is proved that the image of $x$ in the irreducible representation of $G$ with highest weight $\omega$ contains more than $d(r-m)$ Jordan blocks of size $p$ if $m$ and $r-m$ are not too small and $\sigma _x(\omega )\geq p-1+c_x$.

References

P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups. I, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 3, 401–425. MR 417306, DOI 10.1017/S0305004100052403
N. Bourbaki, Groupes et algebres de Lie. Ch. IV–VI, Hermann, Paris, 1968.
N. Bourbaki, Groupes et algebres de Lie. Ch. VII–VIII, Hermann, Paris, 1975.
Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
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, DOI 10.1017/CBO9780511629235
R. Lawther and D. M. Testerman, $A_1$ subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 141 (1999), no. 674, viii+131. MR 1466951, DOI 10.1090/memo/0674
Gary M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286. MR 888704, DOI 10.1090/memo/0365
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
Robert Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56. MR 155937, DOI 10.1017/S0027763000011016
Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
I.D. Suprunenko, The minimal polynomials of elements of order $p$ in irreducible representations of Chevalley groups over fields of characteristic $p$, Siberian Adv. Math. 6 (1996), no 4, 97–150. Translated from Problems of Algebra and Logic. Trudy Inst. Math. Sib. Otdel. RAN. Novosibirsk. Vol. 30 (1996), 126–163.
Irina D. Suprunenko, On Jordan blocks of elements of order $p$ in irreducible representations of classical groups with $p$-large highest weights, J. Algebra 191 (1997), no. 2, 589–627. MR 1448810, DOI 10.1006/jabr.1996.6916
I.D. Suprunenko, $p$-large representations and asymptotics: a survey and conjectures, Algebra 12. J. Math. Sci. (New York) 100 (2000), no. 1, 1861–1870.
Donna M. Testerman, $A_1$-type overgroups of elements of order $p$ in semisimple algebraic groups and the associated finite groups, J. Algebra 177 (1995), no. 1, 34–76. MR 1356359, DOI 10.1006/jabr.1995.1285