 If λ is a diagram where each row has no more than "k" cells, then Sλ("V") is an irreducible GL("V")-representation of highest weight λ.In fact, any rational representation of GL("V") is isomorphic to a direct sum of representations of the form Sλ("V") ⊗ det("V")⊗"m", where λ is a Young diagram with each row strictly shorter than "k", and "m" is any (possibly negative) integer.In this context Schur-Weyl duality states that as a formula_21-module

where formula_23 is the number of standard young tableaux of shape λ.