Broué’s abelian defect group conjecture

A list of vector spaces does not tell us how maps connect them. A map between two one-dimensional spaces can pass every vector through unchanged or send every vector to zero, despite using spaces of the same size. A representation block groups related actions of a finite group on vector spaces. Broué asks whether, when its defect group has commuting elements, a block for the whole group and a corresponding block for the subgroup preserving that defect group always give equivalent ways to study chains of representation spaces and their maps. The one-map example explains why maps matter; it is not a finite-group block example.[1] [2]

Consider two one-dimensional vector spaces linked by a map, forming a two-term complex. The identity map gives zero homology in both degrees; the zero map gives one-dimensional homology in both. The same vector-space sizes therefore leave important information undetermined. For representation blocks, Broué’s conjecture asks whether the complexes and their homological relationships for a whole-group block always have a local counterpart when the defect group is abelian.

Let k be algebraically closed of characteristic p, B a block of kG with abelian defect group D, and b its Brauer correspondent in $kN_G(D)$. Modules are vector spaces carrying the relevant algebra action; a bounded complex is a finite chain of them with consecutive maps composing to zero. The classical target is an equivalence $D^b(B\text{-mod})\simeq D^b(b\text{-mod})$ of their bounded derived categories. This does not assert an equivalence of the ordinary module categories. Splendid Rickard and coefficient-descent refinements supply additional structure in studied cases; the question here is the classical assertion.[1] [2]

This correspondence is known for many families. It is still unproved for every block satisfying the abelian-defect condition. Matching the number of irreducible representations would be much less information than the conjecture requires.[1] [2]

Zhou–Zhang prove it for 2-blocks with abelian defect and prime-order inertial quotient, not all 2-blocks. Du–Huang’s August 2026 work proves descent for specified double-cover RoCK blocks. Already proved symmetric-group cases are progress, not an unresolved subcase; arbitrary remaining abelian-defect blocks lack a universal construction.[1] [2]

Knowing only the number of basic group actions misses how larger actions and chains of them fit together. The conjectured local description would let the whole-group block and its local counterpart be used to study the same homological questions. The identity-versus-zero example shows why those connections carry information that a numerical count leaves out.[1] [2]

A derived equivalence supplies a translation between two settings for homological questions, including the maps among their complexes, even when their ordinary module categories differ. It would explain how the subgroup stabilizing the defect group captures this aspect of a global representation algebra. The elementary identity-versus-zero complex shows why preserving connections is consequential: identical component sizes do not determine homology. The issue is the validity of that local description across abelian-defect blocks, without extending it to arbitrary defect groups.[1] [2]