We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behaviour of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC). We describe the orbit space of the SLOCC group SL(3, C ) ×3 both in its affine and projective versions in terms of a very symmetric normal form parameterized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals.The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes.

In the late 1930s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.

International audienc

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International audience We show that the Kronecker coefficients indexed by two two―row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients. Nous démontrons que les coefficients de Kronecker indexés par deux partitions de longueur au plus 2 sont donnés par des formules quasipolynomiales quadratiques dont les domaines de validité sont les cellules maximales d'un éventail. Des calculs simples nous donnent une description explicite des formules quasipolynomiales et de l'éventail associé. Ces nouvelles formulas sont obtenues de formules analogues pour les coefficients de Kronecker réduits correspondants et au moyen d'une formule reconstruisant les coefficients de Kronecker à partir des coefficients de Kronecker réduits. Une application est la caractérisation exacte de tous les coefficients de Kronecker non―nuls indexés par deux partitions de longueur au plus deux. Ceci nous a permis de réfuter une conjecture de Mulmuley au sujet des fonctions de dilatations associées aux coefficients de Kronecker.