We prove a conjecture of Stembridge concerning stability of Kronecker coefficients that vastly generalizes Murnaghan's theorem. The main idea is to identify the sequences of Kronecker coefficients in question with Hilbert functions of modules over finitely generated algebras. The proof only uses Schur---Weyl duality and the Borel---Weil theorem and does not rely on any existing work on Kronecker coefficients.
Vallejo introduces a notion of additive stability in @cite_5 and proves that it implies stability in Stembridge's sense. Additive stability is provided by the existence of a certain additive matrix, and hence is easier to apply, but it is less general [Example 6.3] vallejo .
@cite_5
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In this paper we give a new sufficient condition for a general stability of Kronecker coefficients, which we call it additive stability. It was motivated by a recent talk of J. Stembridge at the conference in honor of Richard P. Stanley's 70th birthday, and it is based on work of the author on discrete tomography along the years. The main contribution of this paper is the discovery of the connection between additivity of integer matrices and stability of Kronecker coefficients. Additivity, in our context, is a concept from discrete tomography. Its advantage is that it is very easy to produce lots of examples of additive matrices and therefore of new instances of stability properties. We also show that Stembridge's hypothesis and additivity are closely related, and prove that all stability properties of Kronecker coefficients discovered before fit into additive stability.