Abstract
Communicated by Academician Vyacheslav I. Yanchevskii In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characteristics are found. It is proved that if p > 5 for a group of type E 8 and p > 3 for other exceptional algebraic groups, then for irreducible representations of these groups in characteristic p with large highest weights with respect to p, the degree of the minimal polynomial of the image of a unipotent element is equal to the order of this element.
Highlights
In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characte ristics are found
Doklady of the National Academy of Sciences of Belarus, 2019, vol 63, no. 5, pp. 519–525 respect to p, the degree of the minimal polynomial of the image of a unipotent element is equal to the order of this element
Throughout the text C is the complex field, K is an algebraically closed field of an odd characteristic p, Z and Z+ are the sets of integers and nonnegative integers, respectively, G is a connected simple algebraic group of an exceptional type over K, GC is the connected simple algebraic group over C of the same type as G, r is the rank of G, wi, 1 £ i £ r, are the fundamental weights of G, w(j) is the highest weight of a representation j
Summary
In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characte ristics are found. Let p = 3, x Î G be a regular unipotent element, and j be a nontrivial irreducible representation of G.
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