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On character tables related to the alternating groups. (English) Zbl 1068.20012

Summary: There is a simple formula for the absolute value of the determinant of the character table of the symmetric group \(S_n\). It equals \(a_{\mathcal P}\), the product of all parts of all partitions of \(n\) [see G. D. James, The representation theory of the symmetric groups. (Lect. Notes Math. 682) (1978; Zbl 0393.20009), Corollary 6.5]. In this paper we calculate the absolute values of the determinants of certain submatrices of the character table \(\mathcal X\) of the alternating group \(A_n\), including that of \(\mathcal X\) itself (Section 2). We also study explicitly the powers of 2 occurring in these determinants using generating functions (Section 3).

MSC:

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory

Citations:

Zbl 0393.20009
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