Ouch! Qiaochu Yuan is correct; W is an irreducible representation of H not G.

I should have said that c can also depend on dim W. As for U(n) I believe the irreducibles do not satisfy lim dim V_n = ∞. There are an infinite number of irreducibles of dimension 1 given by powers of the determinant representation.

Correct. The question is: does there exist an irreducible V such that..

How does that follow?

Yes, H is closed but not necessarily connected.

I am looking for the existence of a uniform bound depending only on n and independent of H.

I think the empty set is considered linearly independent; it amounts to augmenting the simplicial complex. A maximum element would be adding a cone point.

Exactly what I wanted. Thank you.