Geodesic
On the behaviour of small quadratic elements in representations of the special linear group with large highest weights
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 15, Tome 343 (2007), pp. 84-120 
M. V. Velichko; I. D. Suprunenko. On the behaviour of small quadratic elements in representations of the special linear group with large highest weights. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 15, Tome 343 (2007), pp. 84-120. http://geodesic.mathdoc.fr/item/ZNSL_2007_343_a3/
@article{ZNSL_2007_343_a3,
     author = {M. V. Velichko and I. D. Suprunenko},
     title = {On the behaviour of small quadratic elements in representations of the special linear group with large highest weights},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {84--120},
     year = {2007},
     volume = {343},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_343_a3/}
}
TY  - JOUR
AU  - M. V. Velichko
AU  - I. D. Suprunenko
TI  - On the behaviour of small quadratic elements in representations of the special linear group with large highest weights
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2007
SP  - 84
EP  - 120
VL  - 343
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2007_343_a3/
LA  - ru
ID  - ZNSL_2007_343_a3
ER  - 
%0 Journal Article
%A M. V. Velichko
%A I. D. Suprunenko
%T On the behaviour of small quadratic elements in representations of the special linear group with large highest weights
%J Zapiski Nauchnykh Seminarov POMI
%D 2007
%P 84-120
%V 343
%U http://geodesic.mathdoc.fr/item/ZNSL_2007_343_a3/
%G ru
%F ZNSL_2007_343_a3

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

For almost all $p$-restricted irreducible representations of the groups $A_n(K)$ in characteristic $p>0$ with highest weights large with respect to $p$ the Jordan block structure of images of small quadratic unipotent elements in these representations is determined. It is proved that if $\varphi$ is an irreducible $p$-restricted representation of $A_n(K)$ in characteristic $p>0$ with highest weight $$ m_1\omega_1+\ldots+m_n\omega_n, \quad \sum_{i=1}^n m_i\ge p-1, $$ not too few of the coefficients $m_i$ are less than $p-1$ and $n$ is large enough with respect to the codimension of the fixed subspace of an element $z$ under consideration, then $\varphi(z)$ has blocks of all sizes from 1 to $p$.

[1] N. Burbaki, Gruppy i algebry Li, Gl. IV–VI, Mir., M., 1972 | MR | Zbl

[2] N. Burbaki, Gruppy i algebry Li, VII–VIII, Mir., M., 1978 | MR

[3] M. V. Velichko, “O povedenii kornevykh elementov v modulyarnykh predstavleniyakh simplekticheskikh grupp”, Tr. IM NAN Belarusi, 14:2 (2006), 28–34

[4] M. V. Velichko, “O blochnoi strukture dlinnykh kornevykh elementov v neprivodimykh predstavleniyakh algebraicheskikh grupp tipa $C_n$”, Vestnik molodykh uchenykh MGU “Lomonosov”, 2006, no. III, 232–235

[5] R. Steinberg, Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[6] I. D. Suprunenko,, “Minimalnye polinomy elementov poryadka $p$ v neprivodimykh predstavleniyakh grupp Shevalle nad polyami kharakteristiki $p$”, Problemy algebry i logiki, Tr. IM SO RAN, 30, 1996, 126–163 | Zbl

[7] U. Feit, Teoriya predstavlenii konechnykh grupp, Nauka, M., 1990 | MR | Zbl

[8] J. C. Jantzen, “Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen”, Bonner Math. Schr., 67, 1973 | MR | Zbl

[9] A. A. Osinovskaya, I. D. Suprunenko, “On the Jordan block structure of images of some unipotent elements in modular irredcible representations of the classical algebraic groups”, J. Algebra, 273 (2004), 586–600 | DOI | MR | Zbl

[10] S. Smith, “Irreducible modules and parabolic subgroups”, J. Algebra, 75 (1982), 286–289 | DOI | MR | Zbl

[11] I. D. Suprunenko, “On Jordan blocks of elements of order $p$ in irreducible representations of classical groups with $p$-large highest weights”, J. Algebra, 191 (1997), 589–627 | DOI | MR | Zbl

[12] I. D. Suprunenko, “The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic in Mem. Amer. Math. Soc.” (to appear) | MR

[13] P. H. Tiep, A. E. Zalesskii, “Mod $p$ reducibility of unramified representations of finite groups of Lie type”, Proc. London Math. Soc., 84 (2002), 439–472 | DOI | MR | Zbl

[14] M. V. Velichko, “On the behaviour of root elements in irreducible representations of simple algebraic groups”, Tr. IM NAN Belarusi, 13:2 (2005), 116–121

[15] A. A. Osinovskaya, “Restrictions of representations of algebraic group of types $E_n$ and $F_4$ to naturally embedded $A_1$-subgroups and the behavior of root elements”, Comm. Algebra, 33 (2005), 213–220 | DOI | MR | Zbl