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The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic
I. D. Suprunenko, National Academy of Sciences of Belarus, Minsk, Belarus
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Memoirs of the American Mathematical Society
2009; 154 pp; softcover
Volume: 200
ISBN-10: 0-8218-4369-9
ISBN-13: 978-0-8218-4369-7
List Price: US$72 Individual Members: US$43.20
Institutional Members: US\$57.60
Order Code: MEMO/200/939

The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form $$(t-1)^d$$ and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated.

In the general case the problem of computing the minimal polynomial of the image of a given element of order $$p^s$$ in a fixed irreducible representation of a classical group over a field of characteristic $$p>2$$ can be reduced to a similar problem for certain $$s$$ unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I-XII.

The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from $$p$$-solvable linear groups to semisimple groups.

• Introduction
• Notation and preliminary facts
• The general scheme of the proof of the main results
• $$p$$-large representations
• Regular unipotent elements for $$n=p^s+b$$, $$0 < b < p$$
• A special case for $$G=B_r(K)$$
• The exceptional cases in Theorem 1.7
• Theorem 1.9 for regular unipotent elements and groups of types $$A$$, $$B$$, and $$C$$
• The general case for regular elements
• Theorem 1.3 for groups of types $$A_r$$ and $$B_r$$ and regular elements
• Proofs of the main theorems
• Some examples
• Appendix. Tables
• Appendix. Bibliography
• Appendix. Index