9 D ec 1 99 7 Cycle Indices for the Finite Classical Groups By Jason
نویسنده
چکیده
The Polya cycle index has been a key tool in understanding what a typical permutation π ∈ Sn ”looks like”. It is useful for studying properties of a permutation which depend only on its cycle structure. Here are some examples of theorems which can be proved using the cycle index. Lloyd and Shepp [25] showed that for any i < ∞, the joint distribution of (a1(π), · · · , ai(π)) for π chosen uniformly in Sn converges to independent (Poisson(1), · · ·, Poisson(1i )) as n → ∞. Goncharov [17] proved that the number of cycles in a random permutation is asymptotically normal with mean log n and standard deviation (log n) 1 2 . Goh and Schmutz [14] proved that if μn is the average order of an element of Sn, then:
منابع مشابه
9 D ec 1 99 7 Cycle Indices for the Finite Classical
This paper deenes and develops cycle indices for the nite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.
متن کاملCycle Indices for the Finite Classical Groups Running Title: Cycle Indices for the Nite Classical Groups
This paper de nes and develops cycle indices for the nite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.
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