Multiplicities of Schur functions in invariants of two 3×3 matrices

We relate with any symmetric function presented as an infinite linear combination of Schur functions f(x,y)=∑m(λ1,λ2)S(λ1,λ2)(x,y) the multiplicity series M(f)=∑m(λ1,λ2)tλ1uλ2. We study the behavior of M(f) under natural combinatorial and algebraic constructions. In particular, we calculate the multiplicity series for the symmetric algebra of the irreducible -module corresponding to the complete symmetric function of degree 3. Our main result is that we have found the explicit form of the multiplicity series for the Hilbert (or Poincaré) series of the algebra of invariants of two 3×3 matrices. As a consequence, we have precised the result of Berele on the asymptotics of the multiplicities in the trace cocharacter sequence of two 3×3 matrices.