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

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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 a_{1}ω_{1} +· · ·+a_{r}ω_{r}, this fact is proved when a_{j} ≠ 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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