A Proof of a Generalized Nakayama Conjecture

Abstract

In a recent paper Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general H-analogue of the Nakayama conjecture where H is a set of positive integers.

On generalized blocks for alternating groups

“…Following [8] we say that a permutation (of finite order) is H -regular if for all h ∈ H no cycle (of its disjoint cycle decomposition) has length equal to h. Let G be the symmetric group S n or the alternating group A n , and let H (G) denote the set of H -regular elements of G. For complex irreducible characters α, β of G we define α, β H (G) as in (1). We will prove the following generalization of Theorem 1.1 for d > 1 odd.…”

“…A special case of Theorem 5.2 of [8], and a slight extension of Theorem 5.12 of [6], is the following. If d ∈ H , then for all irreducible characters χ β μ of Z d S w , the algebraic integer…”

“…When d is a prime, then this theorem reduces to Kerber's result, so there is nothing to prove in case d = 2. In fact, most of our argument for the proof of Theorem 1.1 only works for d > 1 odd in which case we are able to prove an even stronger result (see Theorem 2.1) using ideas from [8]. If d is an even integer greater than 2, then there are integers n for which the conclusion of Theorem 1.1 is false.…”

“…In this paper, as in [6] and [8], the 'perfect isometry' will only be applied in the special case when χ β λ is the trivial character 1 of the generalized symmetric group Z d S w . A special case of Theorem 5.2 of [8], and a slight extension of Theorem 5.12 of [6], is the following.…”

Generalized Nakayama Conjecture for C-Orthogonal-Finite Algebras

“…   。Auslander 与 Reiten 合作证 明了有限表示型代数上的正确性。许多学者，如 Yamagata, Fuller 等在一些特殊的代数上做出了贡献 [3][4][5][6] 。现 在它仍然是代数学家们关心的重要公开问题。 2008 年罗和黄 [7] 结合相对同调代数提出了广义 Nakayama 猜想的特殊形式也就是所谓的 Gorenstein 投射猜 想如下：一个自正交的 Gorenstein 投射模 M [8] 是投射的。2010 年，本文作者证明了 Gorenstein 投射猜想对任 意一个 CM-有限代数 [9][10][11][12]…”