Abstract
Sn only by a permutation of its rows and columns, then G is isomorphic to Sn. In this paper we consider a question naturally raised by Nagao’s result. To state it, we must recall that the ordinary irreducible characters of Sn are canonically labelled by the partitions of n, and that this set also labels the conjugacy classes of Sn. Given partitions and of n, let ( ) be the value of the irreducible character of Sn labelled by on elements of cycle type . Now suppose that one has discovered (for example, by applying Nagao’s theorem) that a given square matrix is an unlabelled character table of the symmetric group Sn. We ask: when can one go further, and uniquely reconstruct the partitions labelling its rows and columns? The answer is
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