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

Abstract

Let A be the principal 3-block of a finite group G with an abelian Sylow 3-subgroup P . Let C A be the Cartan matrix of A, and we denote by ρ(C A ) the unique largest eigenvalue of C A . The value ρ(C A ) is called the Frobenius-Perron eigenvalue of C A . We shall prove that ρ(C A ) is a rational number if and only if A and the principal 3-block of N G (P ) are Morita equivalent. This generalizes earlier Wada's theorem in 2007, where he proves it only for the case that the order of P is nine, while we prove it for the case that P is an arbitrary finite abelian 3-group. The result presented here uses the classification of finite simple groups.

A counterexample to a conjecture of Wada

“…In view of Broué's Abelian Defect Group Conjecture, Kiyota, Murai and Wada [3, Question 3] raised the following question. It was proved in [6,5] that the answer to Question 2 is yes for principal p-blocks whenever p ∈ {2, 3}. However, the following counterexample shows not only that Conjecture 1 is false, but also that Question 2 has a negative answer (for principal blocks) in general:…”

“…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]).…”