Conjugacy in Permutation Representations of the Symmetric Group |
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Authors: | Yona Cherniavsky Mishael Sklarz |
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Affiliation: | 1. Department of Mathematics , Bar Ilan University , Ramat Gan, Israel chrnvsk@gmail.com;4. cherniy@math.biu.ac.il;5. Department of Mathematics , Bar Ilan University , Ramat Gan, Israel |
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Abstract: | Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S n × S n . We also consider the permutation representations of S n which arise from the action of S n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not. |
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Keywords: | Characters Conjugacy classes Fixed points Permutation representations Symmetric group |
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