SMALLEST DEGREES OF REPRESENTATIONS OF EXCEPTIONAL GROUPS OF LIE TYPE
SMALLEST DEGREES OF REPRESENTATIONS OF EXCEPTIONAL GROUPS OF LIE TYPE
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特殊李型组的最小表示度
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发表时间：
2001
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通讯作者：
Frank Lübeck
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作者：
Frank Lübeck
其中m∈N， q是素数p的任意幂。对于这个列表中的群G，设G:= G(q)sc是由单连通代数群的Frobenius映射下的不动点群产生的Lie型对应有限群。在有限个例外情况下，G是G和G ~ = G/Z(G)的泛覆盖群。在这种情况下，G的非平凡射影表示的最小度数等于G的非平凡表示的最小度数（G是一个完美群）作为本笔记的主要结果，我们在第2节中确定了群G的复数表示的前几个最小非平凡度，以及它们的多重度。这是delign -Lusztig理论和Lusztig对李型有限群不可约特征的分类的一个应用。具有特殊通用覆盖物的组（以及山雀组F4(2) '）在第3节中列出并处理。完成上述列表中所有组的d0(G)的测定。在第4节中，我们为前五种类型的群收集了l a素数不等于p的一些值dl(G)。在前三种情况下，信息是完整的。这改进了Landazuri， Seitz和Zalesskii在[LS74]和[SZ93]中给出的dl(G), l 6= p的已知下界。
where m ∈ N and q is an arbitrary power of a prime p. For a group G in this list let G := G(q)sc be a corresponding finite group of Lie type arising as group of fixed points under a Frobenius map of a simple simply-connected algebraic group. Up to a finite number of exceptions G is the universal covering group of G and G ∼= G/Z(G). So, in this case, the smallest degrees of non-trivial projective representations of G are equal to the smallest non-trivial degrees of representations of G (which is a perfect group). As the main result of this note we determine in Section 2 the first few smallest non-trivial degrees of complex representations of the groups G, together with their multiplicities. We get these as an application of Deligne-Lusztig theory and Lusztig’s classification of irreducible characters of finite groups of Lie type. The groups with exceptional universal coverings (as well as the Tits group F4(2) ′) are listed and dealt with in Section 3. This completes the determination of d0(G) for all groups in the above list. In Section 4 we collect for the first five types of groups some values dl(G) for l a prime not equal to p. The information is complete in the first three cases. This improves the known lower bounds for dl(G), l 6= p, given by Landazuri, Seitz and Zalesskii in [LS74] and [SZ93].