\(p\)-Steinberg characters of alternating and projective special linear groups (Q1913945)

Every finite group of Lie type in characteristic \(p\) has a Steinberg character, which has value 0 on every \(p\)-singular element, and whose value on every \(p\)-regular element \(x\) is plus or minus the order of the \(p\)-part of the centralizer of \(x\) (in particular, the degree of the character is the order of the Sylow \(p\)-subgroup). Taking these properties as the definition of a \(p\)-Steinberg character of an arbitrary group, W. Feit asks which simple groups have a \(p\)-Steinberg character, and conjectures that all examples are known. In this paper, the author uses the well-developed character theory of the symmetric groups and general linear groups to show that there are no surprises in these cases. In particular, the only \(p\)-Steinberg characters for alternating groups are those arising from the isomorphisms \(A_5\cong L_2(4)\cong L_2(5)\), \(A_6\cong L_2(9)\) and \(A_8\cong L_4(2)\). Similarly, the only \(p\)-Steinberg characters for \(L_n(q)\), where \(p\) is not the defining characteristic, arise from the isomorphisms \(L_2(4)\cong L_2(5)\) and \(L_2(7)\cong L_3(2)\).