A unit group in a character ring of an alternating group (Q1185410)

Let \(R(G)\) be the character ring of a finite group \(G\), let \(U(R(G))\) be the set of all the units in \(R(G)\), and \(U_ f(R(G))\) the set of all the elements of finite orders in \(U(R(G))\). The author shows that \(U(R(G))\) is finitely generated and hence the factor group \(U(R(G))/U_ f(R(G))\) is a free abelian group of finite rank. Now assume that \(G\) is the alternating group \(A_ n\) on \(n\) symbols (\(n>0\)). Denote by \(c(n)\) the number of all self-dual partitions \((m_ 1,m_ 2,\dots,m_ r)\) of \(n\) (i.e. \(m_ i\), \(1 \leq i \leq r\), are integers satisfying \(m_ 1 \geq m_ 2 \geq \cdots \geq m_ r > 0\), \(m_ 1 + m_ 2 + \cdots + m_ r = n\) and \(m_ i = \#\{j \mid m_ j \geq i\}\)) such that the integer \(p = \prod^ k_{i = 1} q_ i\) is not a square number and \(p \equiv 1 \pmod 4\), where the numbers \(q_ 1 > q_ 2 > \cdots > q_ k\) are defined by \(q_ i = 2m_ i - 1\) and \(q_ 1 + q_ 2 + \cdots + q_ k = n\). Then the author shows that the rank of \(U(R(A_ n))/U_ f(R(A_ n))\) is exactly equal to \(c(n)\) for \(n \geq 5\). He also gives an explicit description for the sets \(U(R(A_ k))\) and \(U_ f(R(A_ k))\), \(k = 3, 4\), as well as for the sets \(U(R(S_ n))\) and \(U_ f(R(S_ n))\), \(n > 0\), where \(S_ n\) is the symmetric group on \(n\) symbols.