Regular character tables of symmetric groups.
Let \(X_n\) be the character table of the symmetric group \(S_n\), \(a_n\) the product of all parts of all partitions of \(n\), and \(b_n\) the product of all factorials of multiplicities of parts of partitions of \(n\). It is a well known fact that the absolute value of the determinant of \(X_n\) equals \(a_n\) and also equals \(b_n\); see \textit{F. W. Schmidt} and \textit{R. Simion} [J. Comb. Theory, Ser. A 36, 249-252 (1984; Zbl 0532.05003)]. In this note the author generalizes this result to certain submatrices of \(X_n\), called the regular and singular character tables. For an integer \(l\geq 2\), a partition is called \(l\)-regular if no part is repeated \(l\) or more times, and is called \(l\)-class regular if no part is divisible by \(l\). Partitions which are not (class) regular are called (class) singular. An irreducible character is called regular (singular) if its corresponding partition is regular (singular). We call a conjugacy class regular (singular) if its corresponding partition is class regular (singular). The regular character table \(X_n^{\text{reg}}\) contains the values of regular characters on regular classes and \(X_n^{\text{sing}}\) is defined analogously. The main results are the following: \(|\det(X_n^{\text{reg}})|\) equals the product of all parts of all class regular partitions of \(n\) (when \(l\) is prime, this is also the absolute value of the determinant of the Brauer character table in characteristic \(l\)), and \(|\det(X_n^{\text{sing}})|\) is the product of the factorials of the multiplicities of all parts of class singular partitions of \(n\).
character tables of finite symmetric groups
scientific article; zbMATH DE number 1907106