Projective Modules, Filtrations and Cartan Invariants

Abstract

PROJECT1VE MODULES. FILTRATIONS AND CARTAN INVARIANTS/cG-modules U, K, W, V and an integer / such that p' L 0 V has the same composition factors as V, as do p'L © U' and V. But now p'M © (p'L ® U ) @ p'P ® U ' * (p'L 0 V ) © p'P' © C , and we conclude that M has the desired property.We remark, finally, that a careful examination of this proof will show that we have, in fact, also proved that the Cartan matrix is nonsingular.

Rationality of Eigenvalues of Cartan Matrices in Finite Groups

“…Another example is the case SD(3B) 2 We next show that if ρ B ∈ Z then B and its Brauer correspondent b are Morita equivalent. Suppose that l B = 1.…”

“…However the Cartan matrix of each of the following types is determined only by the defect. There are 6 types, D(3A) 1 , D(3B) 1 , and D(3K), of Cartan matrices of blocks with dihedral defect group, 6 types, l B = 1, Q(2A), Q(2B) 1 , Q(3A) 2 , Q(3B), and Q(3K), with generalized quaternion defect group, and 11 types, l B = 1, SD(2A) 1 , SD(2A) 2 , SD(2B) 1 , SD(2B) 2 , SD(3A) 1 , SD(3B) 1 , SD(3B) 2 , SD(3C) 2 , SD(3D), and SD(3H), with a semidihedral defect group. We should remark that Q(2B) 2 and SD(2B) 3 , where s is a power of 2.…”

Jennings' theorem for p-split groups

“…Suppose also that q(z) = 0. For every two elements v 1 and v 2 of V , define the Lie bracket [v 1 , v 2 ] to be (v 1 , v 2 )z and the "square" v [2] 1 to be q(v 1 )z. This is isomorphic to the 2-restricted Lie algebra created from the grading of an extraspecial 2-group by its dimension subgroups (defined in Section 2).…”

“…Since P k ker(f ),g (1) 1 k g ↑ k ker(f ),g , evaluating Eq. (2) of this paper at t = 1 produces the Brauer character of this principal projective indecomposable module. In a future paper [11], this decomposition of the Brauer character into a product of sums of roots of unity is crucial for determining the asymptotics of the Cartan matrices of the family of Frattini covers, having p-group kernel, of a given group Γ .…”

“…For any finite group Γ and any kΓ -module M, let P kΓ (M) denote the minimal projective cover of M by a kΓ -module. The theorem of Alperin-Collins-Sibley [2] says that, for any kΓ -module M, P kΥ (M) and P kΓ (M) ⊗ D f have the same composition series factors (including multiplicity), where D f is the permutation kΥ -module with basis ker(f ) on which Υ acts by conjugation. Let g ∈ Υ have order prime to p and let C denote the conjugacy class of g in both Υ and ker(f ), g .…”