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Virtual Morita equivalences and Brauer character bijections

Xin Huang

Abstract

We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Brauer character bijections between blocks to virtual Morita equivalences. As a corollary, we obtain that Navarro's refinement of Alperin's weight conjecture holds for blocks with cyclic and Klein four defect groups, blocks of symmetric and alternating groups with abelian defect groups, and $p$-blocks of ${\rm SL}_2(q)$ and ${\rm GL}_2(q)$, where $p|q$.

Virtual Morita equivalences and Brauer character bijections

Abstract

We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Brauer character bijections between blocks to virtual Morita equivalences. As a corollary, we obtain that Navarro's refinement of Alperin's weight conjecture holds for blocks with cyclic and Klein four defect groups, blocks of symmetric and alternating groups with abelian defect groups, and -blocks of and , where .
Paper Structure (2 theorems, 1 equation)

This paper contains 2 theorems, 1 equation.

Key Result

Theorem 1

Let $G$ and $H$ be finite groups and let $n$ be a positive integer large enough for $G$ and $H$. Let $b$ and $c$ be block idempotents of $\mathcal{O} G$ and $\mathcal{O} H$ respectively. Suppose that $\mathcal{O}$ is absolutely unramified (i.e. $J(\mathcal{O})=p\mathcal{O}$). If there is a virtual M

Theorems & Definitions (4)

  • Theorem 1
  • Conjecture 2
  • Corollary 3
  • Remark 4