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=1rCni under the action of U(n1)×⋯×U(nr), 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 ith particle has ni 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.