On the characteristic map of finite unitary groups
In his classic book on symmetric functions, Macdonald describes a remarkable; result by Green relating the character theory of the finite general linear; group to transition matrices between bases of symmetric functions. This; connection allows us to analyze the representation theory of the general linear; group via symmetric group combinatorics. Using the work of Ennola, Kawanaka,; Lusztig and Srinivasan, this paper describes the analogous setting for the; finite unitary group. In particular, we explain the connection between; Deligne-Lusztig theory and Ennola's efforts to generalize Green's work, and; deduce various representation theoretic results from these results.; Applications include finding certain sums of character degrees, and a model of; Deligne-Lusztig type for the finite unitary group, which parallels results of; Klyachko and Inglis and Saxl for the finite general linear group.