# 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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## References

- 1.N. Bourbaki,
*Lie Groups and Algebras*[Russian translation], Chaps. IV–VI, Mir, Moscow (1972).Google Scholar - 2.N. Bourbaki,
*Lie Groups and Algebras*[Russian translation], Chaps. VII–VIII, Mir, Moscow (1978).Google Scholar - 3.M. V. Velichko, “On the behavior of root elements in modular representations of the symplectic group,”
*Trudy Inst. Mat.*,**14**, No. 2, 28–34 (2006).Google Scholar - 4.M. V. Velichko, “Properties of small unipotent elements in modular representations of classical algebraic groups,” Ph.D. Thesis, Minsk (2007).Google Scholar
- 5.M. V. Velichko and I. D. Suprunenko, “Small quadratic elements in representations of the special linear group with large highest weights,”
*Zap. Nauchn. Semin. POMI*,**343**, 84–120 (2007).MathSciNetMATHGoogle Scholar - 6.A. A. Osinovskaya, “Regular unipotent elements from the naturally embedded subgroups of rank 2 in modular representations of classical group,”
*Zap. Nauchn. Semin. POMI*,**356**, 159–178 (2008).MathSciNetGoogle Scholar - 7.A. A. Osinovskaya, “Regular unipotent elements from subsystem subgroups of type
*C*_{2}in representations,”*Tr. Inst. Mat.*,**17**, No. 1, 119–126 (2009).MathSciNetMATHGoogle Scholar - 8.A. A. Osinovskaya and I. D. Suprunenko, “Jordan block structure of unipotent elements from naturally embedded subgroups of type
*A*_{3}in special modular representations of groups of type*A*_{n},”*Dokl. NAN Belarus*,**51**, No. 6, 25–29 (2007).Google Scholar - 9.R. Steinberg,
*Lectures on Chevalley Groups*[Russian translation], Mir, Moscow (1975).Google Scholar - 10.I. D. Suprunenko, “Minimal polynomials of elements of order
*p*in irreducible representations of Chevalley groups over fields of characteristic*p*,” in:*Problems in Algebra and Logic*, Novosibirsk (1996), pp. 126–163.Google Scholar - 11.I. D. Suprunenko, “On the block structure of regular unipotent elements from subsystem subgroups of type
*A*_{1}*× A*_{2}in representations of the special linear group,”*Zap. Nauchn. Semin. POMI*,**388**, 247–269 (2011).MathSciNetGoogle Scholar - 12.W. Feit,
*Representation Theory of Finite Groups*[Russian translation], Nauka, Moscow (1990).MATHGoogle Scholar - 13.R. Lawther, “Jordan block sizes of unipotent elements in exceptional algebraic groups,”
*Commun. Algebra*,**25**, 4125–4156 (1995).MathSciNetCrossRefMATHGoogle Scholar - 14.A. A. Osinovskaya and I. D. Suprunenko, “On the Jordan block structure of images of some unipotent elements in modular irreducible representations of the classical algebraic groups,”
*J. Algebra*,**273**, 586–600 (2004).MathSciNetCrossRefMATHGoogle Scholar - 15.A. A. Osinovskaya, “Restrictions of representations of algebraic groups of types
*E*_{n}and*F*_{4}to naturally embedded*A*_{1}-subgroups and the behavior of root elements,”*Commun. Algebra*,**33**, 213–220 (2005).MathSciNetCrossRefMATHGoogle Scholar - 16.S. Smith, “Irreducible modules and parabolic subgroups,”
*J. Algebra*,**75**, 286–289 (1982).MathSciNetCrossRefMATHGoogle Scholar - 17.I. D. Suprunenko, “The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic,”
*Memoirs Amer. Math. Soc.*,**200**, No. 939 (2009).Google Scholar - 18.P. H. Tiep and A. E. Zalesskii, “Mod
*p*reducibility of unramified representations of finite groups of Lie type,”*Proc. London Math. Soc.*,**84**, 439–472 (2002).MathSciNetCrossRefMATHGoogle Scholar - 19.M. V. Velichko, “On the behavior of the root elements in irreducible representations of simple algebraic groups,”
*Trudy Inst. Mat.*,**13**, No. 2, 116–121 (2005).Google Scholar