Geodesic
On the Jordan block structure of a product of long and short root elements in irreducible representations of algebraic groups of type $B_r$
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 18-31 Cet article a éte moissonné depuis la source Math-Net.Ru

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The behaviour 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 $a_1\omega_1+\dots+a_r\omega_r$ this fact is proved when $a_j\neq p-1$ for some $j and one of the following holds: 1) $a_r\neq p-1$ and $\sum_{i=1}^{r-2}a_i\geq p-1$; 2) $2a_{r-1}+a_r, $\sum_{i=1}^{r-3}a_i\neq0$ for $2a_{r-1}+a_r=p-2$ or $p-1$ and $\sum_{i=1}^{r-3}a_i\neq0$ or $(r-3)(p-1)$ for $a_r=p-1$. 
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     title = {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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T. S. Busel. On the Jordan block structure of a product of long and short root elements in irreducible representations of algebraic groups of type $B_r$. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 18-31. http://geodesic.mathdoc.fr/item/ZNSL_2014_430_a2/

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