The McKay numbers of a subgroup of \(\text{GL}(n;q)\) containing \(\text{SL}(n;q)\)
For a finite group \(G\), a prime number \(p\), a nonnegative integer \(k\) and a \(p\)-block \(B\) of \(G\), let \[ m_p(k,G,B)=|\{\xi\in\text{Irr}(G)\mid v(\xi(1))=k,\;\xi\in B\}|, \] where \(\text{Irr}(G)\) is the set of irreducible complex characters of \(G\) and \(v\) is the exponential valuation of some splitting field of \(G\) with \(v(p)=1\). The sum \(m_p(k,G)=\sum_B m_p(k,G,B)\) over all \(p\)-blocks of \(G\) is called the \(k\)th McKay number of \(G\). Write \[ L_h=L_h(n,q)=\{x\in\text{GL}(n,q)\mid\text{det}(x)\in U_h\}, \] where \(U_h\) is the subgroup of the multiplicative group \(F_1\) of \(\text{GF}(q)\) of order \(h\). Then \[ \text{GL}(n,q)\trianglerighteq L_h(n,q)\trianglerighteq\text{SL}(n,q). \] Denote by \(\text{PL}_h(n,q)\) the factor group of \(L_h(n,q)\) modulo its center \(Z(L_h(n,q))\). The paper under review is concerned with the groups \(L_h(n,q)\) and \(\text{PL}(n,q)\). The number \(m_p(k,G,B)\) is written concretely in terms of several invariants of partitions in the case when \(G=L_h(n,q)\) and \(G=\text{PL}_h(n,q)\). It is also shown that the Alperin-McKay conjecture [\textit{J. L. Alperin}, Proc. Conf. finite groups, Park City 1975, 341-356 (1976; Zbl 0366.20012)] holds for both \(L_h(n,q)\) and \(\text{PL}_h(n,q)\).
scientific article; zbMATH DE number 1270308