Establishing irreduciblity of representations via Gram determinants

This talk is the outcome of my efforts to unify different techniques for establishing ireducibility of unitary representations of infinite-dimensional groups (see my book published  by the EMS in 2018 as Tracts in Mathematics 29).

 It is well known that the distance of the vector from the hyperplane  denerated by n vectors is expressed in terms of the Gram determinants.  If the vectors are in  general positions, the problem is too hard to solve explicitly, since then we should  calculate arbitrary determinants. We present a technique allowing one to calculate explicitly the distance in some particular cases.