Eigenvalues of Cartan matrices of principal 2-blocks with abelian defect groups

Abstract

Let G be a finite group with an abelian Sylow 2-subgroup P . Let C B be the Cartan matrix of the principal 2-block B of G. We show that the Frobenius-Perron eigenvalue ρ(B) of C B is a rational integer if and only if B and its Brauer correspondent block b of N G (P ) are Morita equivalent by using a classification of finite simple groups with an abelian Sylow 2-subgroup. In this case, we can take the Brauer character table Φ b of b as a unimodular eigenvector matrix U B of C B over a complete discrete valuation ring R.

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

“…This generalizes earlier Wada's result [32,Theorem C], where he proves 1.1 and 1.2 for the case where P is of order 9 while we prove the same for an arbitrary finite abelian 3-group P . Of course we were inspired by the work of Kunugi and Wada [21], where they prove the same result for principal 2-blocks of finite groups with abelian Sylow 2-subgroups.…”

“…For instance, 1.1 is true for the cases that P is cyclic, that the corresponding block algebra A over a field is of tame representation type, that A is the principal 3-block and P is elementary abelian of order 9, and that A is the principal 2-block and P is abelian, see [12,13,31,32,21]. Suppose, furthermore, that a p-block A of G has an abelian defect group P .…”

“…We note that 1.2 has a negative answer if P is nonabelian, see [13,Example]. However, there are several positive answers to 1.2, such as the cases that P is cyclic, that A is of tame representation type, that A is the principal 3-block with an elementary abelian defect group of order 9, and that A is the principal 2-block and P is abelian, see [12,13,31,32,21].…”

“…Strategy 1.6. Our procedure is similar to that Kunugi and Wada in [21] take. But, even in such a strategy, we actually had very serious difficulties to reduce to the simple group cases.…”

“…Hence it turns out that C A has the largest eigenvalue with multiplicity one, which is called the Frobenius-Perron eigenvalue and denoted by ρ(C A ). There are several results on the subject such as [12,13,31,32,21], which are due to M. Kiyota, T. Wada, M. Murai and N. Kunugi. They are interested in studying integrality of eigenvalues of C A .…”