On the block structure of regular unipotent elements from subsystem subgroups of type $A_1\times A_2$ in representations of the special linear group

I. D. Suprunenko

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 21, Tome 388 (2011), pp. 247-269

Résumé

We study behavior of regular unipotent elements from a subsystem subgroup of type $A_1\times A_2$ in $p$-restricted irreducible representations of a special linear group of rank greater than $5$ over a field of characteristic $p>2$. For a certain class of such representations with locally small highest weights it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. In particular, the following is proved. Let $K$ be an algebraically closed field of characteristic $p$, $G=A_r(K)$, $r\geq9$, $x\in G$ be a regular unipotent element from a subsystem subgroup of type $A_1\times A_2$, and let $\varphi$ be a $p$-restricted representation of $G$ with highest weight $\sum^r_{j=1}a_j\omega_j$. Set $l=\min\{p,1+2a_1+3(a_2+\dots+a_{r-1})+2a_r\}$. Assume that more than $6$ coefficients $a_j$ are not equal to $p-1$ and that for some $i$, the sum $a_i+a_{i+1}$ for $p>3$ and $a_i=a_{i+1}=0$ or $1$ for $p=3$. Then the element $\varphi(x)$ has Jordan blocks of all sizes from $1$ to $l$.