# Young Tableau Projectors: Does the order of symmetric and anti-symmetric projectors matter?

Given a Young Tableau we find the irreducible basis of an arbitrary tensor by projecting, The projectors are usually defined as first symmetrise over the row entries and then anti-symmetrise over the column entries.

What happens if we interchange this order that is anti-symmetrise first and symmetrise after. I understand that these projectors do not commute, however, do they correspond for a given young tableau, to the same invariant subspace, albeit with different basis?