Some generalized characters associated to a transitive permutation group

Let G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

Permutation characters

We suppose throughout that G is a finite group with a faithful matrix representation X over the complex field. We suppose that X affords a character π of degree r whose values are rational (hence rational integers). If the matrices in some representation of G affording a character π0 are all permutation matrices, then π0 is called a permutation character. Permutation characters have non-negative integral values. In the general case, we consider what properties of permutation characters are true of π, and in particular, under what circumstances π is a permutation character. Note that assuming X to b faithful is equivalent to considering the image group X(G) instead of G.

The characters of the cubic surface group

The characters of the cubic surface group G are got by using its representation as a group of orthogonal projectivities in a four-dimensional space a over GF (3). In α , G permutes 27 pentagons transitively and 6 other characters, in addition to the permutation character of degree 27, are obtained by noting how the 5 vertices of each invariant pentagon are permuted. Other geometrical objects that afford transitive permutation representations are available, and provide material for similar investigations. The papers that have preceded this, wherein not only is the geometry described in detail but also many 5-rowed matrices of the orthogonal group are explicitly given, are laid under contribution. In particular, the table at the end of the most recent paper provides at sight not only permutation characters but monomial characters as well.