The squaring operation on -generators of the Dickson algebra

AbstractWe study the squaring operation Sq0 on the dual of the minimal -generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes–Zarati homomorphism vanishes (1) on every element in any finite Sq0-family in $Ext_{\cala}^*(\Fd, \Fd)$ except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.