The Jordan block structure of unipotent elements in Weyl modules for groups of type $A_1$ over a field of positive characteristic

T. S. Busel

Trudy Instituta matematiki, Tome 21 (2013) no. 2, pp. 70-72

Résumé

The Jordan block structure of unipotent elements in Weyl modules for an algebraic group of type $A_1$ in a positive characteristic $p$ is described. The following theorem is proved. Theorem. Let $K$ be an algebraically closed field of characteristic $p>0$ and $G=A_1(K).$ Assume that $\lambda=ip+j$ and $i,j\ge0,$ $j$ Then a nontrivial unipotent element of $G$ has $i$ Jordan blocks of size $p$ and one block of size $j+1$ in the Weyl module of $G$ with highest weight $\lambda$. Here the weights of $G$ are naturally identified with the integers. This theorem can be useful for investigating the Jordan block structure of unipotent elements in modular representations of simple algebraic groups and finite Chevalley groups.