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首页> 外文期刊>Memoirs of the American Mathematical Society >The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic
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The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic

机译：奇特征古典群不可约表示中单能元素的极小多项式。

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The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd character-istic are found. These polynomials have the form (t)~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 p5 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 al-gorithm is given. Results of explicit computations for groups of small ranks are contained in Tables IXII. 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.
机译：找到了奇数特征域上古典代数群的不可约理性表示中单能元素图像的最小多项式。这些多项式的形式为（t）〜d，因此完全取决于其次数。在正特性中，这种多项式的次数不能超过相关元素的阶数。对于每个单能元素，如果表示的最高权重相对于地面场特性足够大，则在不可约表示中其最小多项式的次数等于该元素的阶数。另一方面，指出了在每个非平凡表示中最小多项式的次数等于该元素的阶数的单能元素的类。在一般情况下，对于特征为p> 2的场，以经典组的固定不可约表示形式计算阶数p5的给定元素的图像的最小多项式的问题可以简化为某些单势元素的类似问题以及复数域上某个半简单群的某种不可约表示。对于后一个问题，给出了一个明确的算法。表IXII包含了针对小级别小组的显式计算结果。这篇文章可以被认为是对将Hall和Higman（1956）关于最小多项式的基本结果从p可解线性组扩展到半简单组的计划的一种贡献。

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