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Journal of Mathematical Sciences

, Volume 219, Issue 3, pp 346–354 | Cite as

On the Jordan Block Structure of a Product of Long and Short Root Elements in Irreducible Representations of Algebraic Groups of Type B r

  • T. S. Busel
Article
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The behavior of a product of commuting long and short root elements of the group of type B r in p-restricted irreducible representations is investigated. For such representations with certain local properties of highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. For a p-restricted representation with highest weight a1ω1 +· · ·+arωr, this fact is proved when aj ≠ p − 1 for some j < r − 1 and one of the following conditions holds: (1) \( {a}_r\ne p-1\kern0.75em and\kern0.5em {\displaystyle \sum_{i=1}^{r-2}{a}_i\ge p-1;} \) and (2) \( 2{a}_{r-1}+{a}_r<p,{\displaystyle \sum_{i=1}^{r-2}r-3}{a}_i\ne 0\;for\;2{a}_{r-1}+{a}_r=p-2\; or\;p-1\; and\;{\displaystyle \sum_{i=1}^{r-3}{a}_i\ne 0} \) or (r−3) (p-1) for a r = p−1.

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of the National Academy of Sciences of BelarusMinskBelarus

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