Basic characters of the unitriangular group (for arbitrary primes) (Q2782620)

Let \(U_n(q)\) denote the unitriangular group of degree \(n\) over the finite field \(\mathbb{F}_q\) with \(q\) elements. This group consists of all unipotent upper triangular \(n\times n\) matrices with coefficients in \(\mathbb{F}_q\). Assume that \(\Phi(n)=\{(i,j)\mid 1\leq i<j\leq n\}\). A subset \(D\subseteq\Phi(n)\) is called a basic subset if \(|D\cap\{(i,j)\mid i<j\leq n\}|\leq 1\) for all \(1\leq i<n\), and \(|D\cap\{(i,j)\mid 1\leq i<j\}|\leq 1\) for all \(1<j\leq n\).NEWLINENEWLINENEWLINEIn [J. Algebra 175, No. 1, 287-319 (1995; Zbl 0835.20052)], this author proved that every irreducible complex character of \(U_n(q)\) is a constituent of a unique basic character for \(p\geq n\) where \(p\) is the characteristic of the field \(\mathbb{F}_q\). In the paper under review, the author extends this result. More precisely he proves the following result. Theorem 1: Let \(\chi\) be an irreducible character of \(U_n(q)\). Then \(\chi\) is a constituent of a unique basic character of \(U_n(q)\); in other words, there exists a unique basic subset \(D\) of \(\Phi(n)\) and a unique map \(\phi\colon D\to\mathbb{F}_q^*\) such that \(\chi\) is a constituent of \(\xi D(\phi)\) where \(\xi D(\phi)\) is a basic character of \(U_n(q)\).