The principal 3-blocks of the 3-dimensional projective special unitary groups in non-defining characteristic (Q2744706)

The main result of the paper is the following Theorem. Let \(G\) be the projective special unitary group \(\text{PSU}(3,q^2)\) for a power \(q\) of a prime such that \(q\equiv 2\) or \(5\pmod 9\), so that a Sylow \(3\)-subgroup \(P\) of \(G\) is elementary Abelian of order \(9\). Let \(H:=N_G(P)\) and let \((K,{\mathcal O},k)\) be a splitting \(3\)-modular system for all subgroups of \(G\). Then the principal block \(\widehat A\) of \({\mathcal O}G\) is Morita equivalent to the principal block \(\widehat B\) of \({\mathcal O}H\).NEWLINENEWLINENEWLINENote that in the situation of the Theorem the group \(H\) is a split extension of \(P\) by a quaternion group of order \(8\) acting faithfully on \(P\); in particular, the principal block of \({\mathcal O}H\) is \({\mathcal O}H\) itself. By the assertion of the Theorem, the principal block \(\widehat A\) of \({\mathcal O}G\) is Morita equivalent to the principal block \(\widehat B\) of \({\mathcal O}H\) which is \({\mathcal O}H\); consequently, \(\widehat A\) and \(\widehat B\) are derived equivalent, and hence so are \(A\) and \(B\), where \(A\) and \(B\) are the principal blocks of \(kG\) and \(kH\), respectively.NEWLINENEWLINENEWLINEWith their result the authors verify the well-known and important conjecture of M. Broué for a special instance. This conjecture says that the principal \(p\)-blocks of \(G\) and \(N_G(P)\) are derived equivalent whenever the Sylow \(p\)-subgroup \(P\) of \(G\) is Abelian.