On the Frobenius–perron Eigenvalues of Cartan Matrices for Some Finite Groups

Abstract

We study integrality of the Frobenius–Perron eigenvalues of the Cartan matrices for the principal blocks of some finite groups of Lie type with noncyclic abelian Sylow p-subgroups.

Eigenvalues of Cartan matrices of principal 3-blocks of finite groups with abelian Sylow 3-subgroups

“…Additionally, 2.1 is powerful too, which is also new and due to Okuyama and Wada [25]. Finally, a recent result of the second author [33] works quite well, too.…”

A counterexample to a conjecture of Wada

“…This has been done for blocks of finite or tame representation type (see [3,Propositions 3 and 4]). For p-solvable G we have (1) ⇔ (2) ⇔ (4) and ρ(C) ≤ |D| (see [ [5,6,9,12]. If D G, then (1)-(4) are satisfied (see [3,Proposition 2]).…”

“…For p-solvable G we have (1) ⇔ (2) ⇔ (4) and ρ(C) ≤ |D| (see [3,Theorem 1], [8,Corollary 3.6] and [4,Corollary 3.6]). Other special cases were considered in [5,6,9,12]. If D G, then (1)-( 4) are satisfied (see [3,Proposition 2]).…”