Dimensions of components of tensor products of the linear groups representations with applications to Beurling-Fourier algebras

We give universal upper bounds on the relative dimensions of isotypic components of a tensor product of the linear group GL(n) representations and universal upper bounds on the relative dimensions of irreducible components of a tensor product of the special linear group SL(n) representations. This problem is motivated by harmonic analysis problems, and we give some applications of this result in the theory of Beurling-Fourier algebras.