Basic sets in defining characteristic for general linear groups of small rank.
Let \(G\) be a finite group, let \(p\) be a prime and let \(\text{Irr}(G)\) be the set of ordinary irreducible characters of \(G\). For \(\chi\in\text{Irr}(G)\), let \(\widehat\chi\) denote the restriction of \(\chi\) to the \(p\)-regular elements of \(G\). A `\(p\)-basic set' of \(G\) is a subset \(B\) of \(\text{Irr}(G)\) such that the set \(\{\widehat\chi\mid\chi\in B\}\) is a \(\mathbb{Z}\)-basis of the \(\mathbb{Z}\)-module generated by \(\{\widehat\chi\mid\chi\in\text{Irr}(G)\}\). Examples show that finding a \(p\)-basic set can help with computing the decomposition matrix of a finite group \(G\). It is not known whether \(p\)-basic sets exist for all finite groups; cases that are known include finite groups of Lie type when \(p\) is different from the defining characteristic. This paper proposes a method for finding a \(p\)-basic set when \(G\) is a finite group of Lie type over a field of characteristic \(p\). Briefly, the method is to find certain ``good relations'' between characters of \(G\) (the author shows how such relations can be generated using Deligne-Lusztig theory and also the Gelfand-Graev character of \(G\)); these relations can then be used to find \(p\)-basic sets in some cases. The method is illustrated via explicit constructions of \(p\)-basic sets for \(G=\text{GL}_n(q)\), where \(q\) is a power of \(p\) and \(n=2\), \(3\) or \(4\).
scientific article; zbMATH DE number 5527491
scientific article