# The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic

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*I. D. Suprunenko*

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.

#### Table of Contents

# Table of Contents

## The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic

- Contents v6 free
- 1. Introduction 18 free
- 2. Notation and preliminary facts 1118 free
- 3. The general scheme of the proof of the main results 3845
- 4. p-large representations 4148
- 5. Regular unipotent elements for n = p[sup(s)] + b, 0 < b < p 5461
- 6. A special case for G = B[sub(r)](K) 6673
- 7. The exceptional cases in Theorem 1.7 7481
- 8. Theorem 1.9 for regular unipotent elements and groups of types A, B, and C 7784
- 9. The general case for regular elements 8188
- 10. Theorem 1.3 for groups of types A[sub(r)] and B[sub(r)] and regular elements 9299
- 11. Proofs of the main theorems 93100
- 12. Some examples 116123
- Appendix. Tables 119126
- Appendix. Bibliography 151158
- Index 153160 free