2010/03/19 by Brian Parshall, Parshall, Brian, Leonard L. Scott +2
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #math.GR
paper · pdf · doi:10.48550/arxiv.1003.3897
The proof of a main result, Theorem 3.3, has been corrected. This correction involved the addition of a part (b) to Lemma 3.2. In an unrelated matter, a new footnote, footnote 11 has been been added, announcing a recently observed new result not in the original paper. Other minor improvements have been made in the exposition
openalex publication_date 2010/03/19 · arxiv created 2011/11/15 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N be a normal subgroup of a group G. An N-module Q is G-stable provided that Q is equivalent to the twist Qg of Q by g, for every g∈ G. If the action of N on Q extends to an action of G on Q, Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q⊕ n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (arXiv:0909.5207v2), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the presentation of a homological obstruction to the existence of a G-module structure, on strongly G-stable modules, and a tensor product approach to killing the obstruction.