Stable Hilbert series as related to the measurement of quantum entanglement

We compute a stable formula for the Hilbert series of the invariant algebra of polynomial functions on ⨂ i = 1 r C n i under the action of U ( n 1 ) × ⋯ × U ( n r ) , when viewed as real vector space. This situation has a physical interpretation as it is the quantum analog of an r -particle classical system in which the i th particle has n i classical states. The stable formula involves only elementary combinatorics, while its derivation involves the representation theory of the symmetric group. In particular, the Kronecker coefficients play an important role.