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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: I. D. Suprunenko
Journal: Proc. Amer. Math. Soc. 129 (2001), 2581-2589
MSC (1991): Primary 20G05; Secondary 20G40
Published electronically: February 9, 2001
MathSciNet review: 1838380
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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 $\sigma_x(\omega)\geq p-1+c_x$.


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Additional Information

I. D. Suprunenko
Affiliation: Institute of Mathematics, National Academy of Sciences of Belarus, Surganov str. 11, Minsk, 220072, Belarus
Email: suprunenko@im.bas-net.by

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05934-2
PII: S 0002-9939(01)05934-2
Keywords: Classical groups, representations, Jordan blocks
Received by editor(s): January 24, 2000
Published electronically: February 9, 2001
Additional Notes: This research has been supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the State program “Mathematical structures” and by the Belarus Basic Research Foundation, Project F98-180.
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society