Formal character tables (Q1074708)

Some well-known arithmetic properties of character tables are formalized in the following Definition 1. A \(k\times k\) matrix X is a formal character table if the following conditions are satisfied: (1) If \(X=(x_{ij})\) then \(x_{1j}=1\) for all j. Next, \(x_{i1}\) is a positive integer for all i. Let \(n=\sum^{k}_{i=1}x^ 2_{i1}\) (n will be referred to as the order of X). Every \(x_{ij}\) is a sum of exactly \(x_{i1}\) nth roots of 1. (2) The algebraic conjugate to any row (resp. column) of X is a row (resp. column). (3) Let \(c_ j=\sum^{k}_{i=1}| x_{ij}|^ 2\); then \(\sum^{k}_{i=1}x_{pi}\bar x_{qi}/c_ i=\delta_{pq}\). (4) The (pointwise) product of any two rows of X is a nonnegative integral combination of the rows of X. \[ (5)\quad a_{pqr}=\frac{n}{c_ pc_ q}\sum^{k}_{s=1}\frac{x_{sp}x_{sq}\bar x_{sr}}{x_{s1}} \] are nonnegative integers for all p,q,r\(\in \{1,...,k\}.\) Let X be a formal character table of order \(n=ds\) where d and s are positive integers and \(s| d\). Define the \((k+1)\times (k+1)\) matrix \[ (X,s)=\left( \begin{matrix} \bar v\quad X\\ d\quad \bar w\end{matrix} \right) \] where \(\bar v\) is the 1st column of X, \(\bar w=(-s,0,...,0)\). Then (X,s) is a formal character table if X is. (X,s) is rarely a character table because of the author's result [Pac. J. Math. 109, 363-385 (1983; Zbl 0536.20005)]. Moreover if (X,s) is a character table of groups then s is a power of a prime p, and if n is the order of X, then \(n=s^ 2(p^{\alpha}-1)\) (Th. 3.1). If X, Y are formal character tables then the Kronecker product \(X\otimes Y\) is a formal character table. Moreover, \(X\otimes Y\) is a character table of a group \(\Leftrightarrow\) X,Y are character tables of groups (Th. 2.4). Let s be a power of a prime p and let \(q=s^ 2\). If X is the character table of SL(2,q), then (X,s) is not the character table of any group (Th. 3.2). The author gives a formal character table (X,2) where X is the character table of \(A_ 5\) and asserts that it is not readily dismissed (X,2) to be a character table. Since the order of (X,2) is 960, (X,2) is a \(6\times 6\) matrix and for a group G with six conjugate classes we have \(| G| \leq 168\) (J. Poland) then we obtain another proof that (X,2) is not a character table.