The Abelian defect group conjecture: some recent progress (Q2759670)

This article surveys recent progress on Broué's Abelian defect group conjecture in block theory (finite group modular representation theory).NEWLINENEWLINENEWLINESection 1 begins with very basic ideas of block theory (Section 1.1) and examples (using \(A_5\) and \(J_1\)) (Section 1.2). Section 1.3 describes derived categories, stable module categories and how the stable module category is equivalent to a natural quotient of the derived category in block theory.NEWLINENEWLINENEWLINELet \(B\) be a block of the finite group \(G\) with defect group \(D\) and let \(b\) be the Brauer correspondent block of \(N_G(D)\) of \(B\). The relationship and similarities between \(B\) and \(b\) is a central issue in block theory.NEWLINENEWLINENEWLINEIn Section 1.4 Broué's conjecture: if \(D\) is Abelian, then the derived categories \(D^b(B)\) and \(D^b(b)\) are equivalent as triangulated categories (Conjecture 1.2) is discussed. This conjecture, with \(D\) Abelian, implies: (1) there is a perfect isometry between \(B\) and \(b\) (a fundamental character theoretic equivalence) and (2) the stable module categories of \(B\) and \(b\) over \(k\) are equivalent as triangulated categories. (Both (1) and (2) are also important conjectures.)NEWLINENEWLINENEWLINESection 1.5 presents perfect isometries in the \(A_5\) and \(J_1\) examples.NEWLINENEWLINENEWLINESection 2: Refinements of the conjecture begins (Section 2.1) with a description of how Morita equivalence suggests sufficient conditions: for (i) a stable module category equivalence (a stable equivalence of Morita type) and (ii) for a derived equivalence.NEWLINENEWLINENEWLINEAssociated to the block \(B\) of \(G\) with defect group \(D\) is a fundamental ``local structure''. Broué posed a ``local structure compatible'' perfect isometry that he called an isotypy: a perfect isometry for each non-trivial subgroup of \(D\) that is ``compatible'' with the global perfect isometry.NEWLINENEWLINENEWLINESection 2.2 describes when \(B\) is the principal block of \(G\) (so that \(D\) is a \(p\)-Sylow subgroup of \(G\)), the fundamental result of the author that gives sufficient conditions on a bimodule complex (called a splendid tilting complex) that yields a ``compatible family'' of derived equivalences and hence an isotypy (Theorem 2.1). This section concludes with the statement of another fundamental result of the author to the effect that a splendid tilting complex over \(k\) can be lifted to \(O\) (Theorem 2.2).NEWLINENEWLINENEWLINESection 2.3 describes a recent partial converse of R. Rouquier: a sufficient condition for producing a complex that induces a global stable equivalence of Morita type for a suitable family of local splendid tilting complexes.NEWLINENEWLINENEWLINESection 2.4 presents applications of Rouquier's procedure to the \(A_5\) and \(J_1\) examples.NEWLINENEWLINENEWLINESection 3 begins with a discussion of recent ideas, results and methods of Linckelmann, Okuyama and the author for lifting a stable equivalence to a derived equivalence and concludes (Section 3.1) with an application to the \(J_1\) example.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].