Let H ⊂ G be reductive groups. Each irreducible representation of G is also a representation of H, but not necessarily irreducible. How does it decompose into irreducibles ? How to describe and compute the multiplicities in its decomposition ? This is the "subgroup restriction problem" for G and H.

In this talk we consider formulas that give the multiplicities for various subroup representation problems in the representation theory of GL_n(C): (i) the Kostka coefficients for C*-diagonal matrices in GL_n(C); (ii) the Littlewood-Richardson coefficients for GL_n(C) ⊂ GL_n(C)xGL_n(C) (diagonal embedding); (iii) the Kronecker coefficients for GL_n(C)xGL_m(C) → GL_mn(C) (Kronecker product of matrices).

We will recall known combinatorial descriptions for the first two families of coefficients (analogous descriptions lack for the Kronecker coefficients). We will also consider formulas studied recently, for the three kinds of coefficients in function of the natural labels of the irreducible representations involved (integer partitions or pairs of partitions). They are of a very special nature, reminiscent of the Ehrhart functions that count integer points inside the dilations of polytopes.