The derived categories of some blocks of symmetric groups and a conjecture of Broué
Let \(p\) be a prime, let \(G\) be a finite group and let \(B\) be a \(p\)-block of \(G\) with defect group \(D\). Also let \(b\) be the \(p\)-block of \(N_G(D)\) that is the Brauer correspondent of \(B\). A well-known conjecture of M. Broué is: If \(D\) is Abelian, then the \(B\)-module and \(b\)-module categories are derived equivalent. A fundamental result of J. Rickard gives necessary and sufficient conditions for the module categories of two rings to be derived equivalent. For \(p\)-blocks \(B\) and \(B'\) of finite groups \(G\) and \(H\) resp., the (left) module categories of \(B\) and \(B'\) are derived equivalent if and only if there exists a bounded complex \(X\) in \(B'\text{-mod-}B\) such that (1) each term of \(X\) is projective in \(B'\text{-mod}\) and in \(\text{mod-}B\); (2) \(X^*\otimes_{B'}X\) is a split complex in \(B\text{-mod-}B\) with homology concentrated in degree \(0\) isomorphic to \(B\); and (3) \(X\otimes_{B}X^*\) is a split complex in \(B'\text{-mod-}B'\) with homology concentrated in degree \(0\) isomorphic to \(B'\). Such a complex is called a Rickard tilting complex. If, in addition, \(G\) and \(H\) have a common \(p\)-subgroup \(D\) which is a defect group of \(B\) and \(B'\) and each term of \(X\) is relatively \(\Delta D\)-projective, then \(X\) is called a splendid Rickard tilting complex and \(B\) and \(B'\) are said to be splendid Rickard equivalent. Broué and Rickard have revised Broué's original conjecture: If \(D\) is Abelian, then the categories \(B'\text{-mod}\) and \(b\text{-mod}\) are splendid Rickard equivalent. The main result of this paper is: If \(p>2\) and \(B\) is a \(p\)-block of a symmetric group with defect group \(D\cong C_p\times C_p\), then \(B\) and its Brauer correspondent \(b\) are splendid Rickard equivalent. Section 2 of this paper presents a proof of an unpublished result of Rickard that all \(p\)-blocks of all symmetric groups with defect group \(D\cong C_p\times C_p\) are derived equivalent. Since A. Marcus has shown that the principal blocks of \((C_p\times C_{p-1})\wr S_2\) and \(S_p\wr S_2\) are derived equivalent, it is necessary to prove that one such block of some symmetric group is derived equivalent to the principal block of \(S_p\wr S_2\). Sections 3-7 are devoted to proving that there is a symmetric group with a defect group \(D\cong C_p\times C_p\) that is Morita equivalent (and hence derived equivalent) to the principal block of \(S_p\wr S_2\). The paper concludes (the main result in Section 8) with a proof that the required derived equivalence is splendid.
scientific article; zbMATH DE number 1340606
scientific article