Let G be a connected, semisimple and simply connected algebraic group and G(q) the corresponding finite Chevalley group over the finite field of order q = p r . In a recent paper the author determined a direct sum decomposition of the kG(q)-submodule generated by a highest weight vector of a certain Weyl module when q is not too small, which is a generalization of Pillen's result in 1997. In this article, we claim that the result does not need the assumption on q.
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.
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.
Abstract. Let G be a connected, semisimple and simply connected algebraic group defined and split over the finite field of order p. Pillen proved in 1997 that the highest weight vectors of some Weyl G-modules generate the principal series modules as submodules for the corresponding finite Chevalley groups. This result is generalized in this paper.
Communicated by Michel Broué Keywords: Broué's abelian defect group conjecture Derived equivalence Tilting complex Stable equivalence of Morita type M. Broué gives an important conjecture which is called Broué's abelian defect group conjecture. This conjecture says that a p-block,where p is a prime number, of a finite group with an abelian defect group is derived equivalent to its Brauer correspondent in the normalizer of the defect group. In this paper, we prove that this conjecture is true for the nonprincipal block of SL(2, p n ) for a positive integer n.
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