Cartan invariants and central ideals of group algebras

Abstract

In this paper we investigate general properties of Cartan invariants of a finite group G in characteristic 2. One of our results shows that the Cartan matrix of G in characteristic 2 contains an odd diagonal entry if and only if G contains a real element of 2-defect zero. We also apply these results to 2-blocks of symmetric groups and to blocks with normal or abelian defect groups. The second part of the paper deals with annihilators of certain ideals in centers of group algebras and blocks.

Cartan invariants and defect zero elements

“…Combining Theorem 3.1 and part (iii) of Theorem 3.3 in [2] we immediately obtain the next result. Here we give an independent and elementary proof which was brought to our attention by Thomas Breuer.…”

“…R e m a r k 2.3. The condition b) in 2.2 does not imply the existence of a p-block of defect zero in general as mentioned already in [2,Section 4]. As an example we may take the alternating group A 7 and p = 2.…”

“…In [2] it has been proved that the condition c ii odd for some i (not necessarily i =ī) is equivalent to the existence of a real 2-defect zero element. The proof relies on subtle computations in the group algebra, and is obtained as a by-product of more general results.…”

“…Exploiting duality arguments in characteristic two we show that the existence of real 2-defect zero elements is equivalent to the existence of odd diagonal entries in the Cartan matrix corresponding to real-valued Brauer characters. In addition, we present an independent and elementary proof of Theorem 3.3 part (iii) in [2].…”

Central ideals and Cartan invariants of symmetric algebras

“…In a sequel [2] to this paper, we will apply our results to group algebras of finite groups. We will see that a finite group G contains a real conjugacy class of 2-defect zero if and only if the Cartan matrix of G in characteristic 2 contains an odd diagonal entry.…”