Kronecker products and the RSK correspondence

The starting point for this work is an identity that relates the number of minimal matrices with prescribed 1-marginals and coefficient sequence to a linear combination of Kronecker coefficients. In this paper we provide a bijection that realizes combinatorially this identity. As a consequence we obtain an algorithm that to each minimal matrix associates a minimal component, with respect to the dominance order, in a Kronecker product, and a combinatorial description of the corresponding Kronecker coefficient in terms of minimal matrices and tableau insertion. Our bijection follows from a generalization of the dual RSK correspondence to 3-dimensional binary matrices, which we state and prove. With the same tools we also obtain a generalization of the RSK correspondence to 3-dimensional integer matrices

Stability of Kronecker coefficients via discrete tomography

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.Comment: 22 page

Stability of Kronecker coefficients via discrete tomography (Extended abstract)

International audienceIn this paper we give a sufficient condition for a general stability of Kronecker coefficients, which we call additive stability. Its main ingredient is the property of a matrix of being additive. This notion seems to be an important one: it appears in Discrete Tomography as a sufficient condition to uniqueness; it also appears in Manivel’s study of asymptotic properties of plethysm through Borel-Weil theory. The proof sketched here combines several results of the author on integer matrices motivated by Discrete Tomography with a new idea of Stembridge, that permits to bound some sequences of Kronecker coefficients. The advantage of additivity with respect to the previous approach by Stembridge is that it is very easy to produce new examples of additive matrices and, therefore, to produce many new examples of stability of Kronecker coefficients. We also show that Murnaghan’s stability property and other instances of stability discovered previously by the author are special cases of additive stability. Besides, our approach permits us to disprove a recent conjecture of Stembridge and to give a new characterization of additivity.Dans ce papier nous donnons une condition suffisant pour la stabilité générale des coefficients de Kronecker, que nous appelons stabilité additive. L'ingrédient principal est la propriété d’une matrice d'être additif. Cette notion est apparemment d’importance: elle apparaît en Tomographie Discrète comme une condition suffisant pour unicité; elle apparaît aussi dans l’étude de Manivel de propriétés asymptotiques du pléthysme par moyen de la théorie de Borel-Weil. La démonstration esquissée ici combine plusieurs résultats de l’auteur sur les matrices à coefficients entiers stimulés pour la Tomographie Discrète avec une nouvelle idée de Stembridge, qui permet de borner quelques successions des coefficients de Kronecker. L’avantage de notre méthode sur l’approche de Stembridge est qu’il est très facile de produire nouveaux exemples de matrices additives, et ainsi, de nouveaux exemples de stabilité des coefficients de Kronecker. Nous démontrons aussi que la stabilité de Murnaghan et d’autres exemples de stabilité trouvés antérieurement par l’auteur sont des cas spéciaux de la stabilité additive. En plus, avec notre approche nous réfutons une conjecture de Stembridge et donnons une nouvelle caractérisation d’additivité

Stability of Kronecker coefficients via discrete tomography (Extended abstract)

In this paper we give a sufficient condition for a general stability of Kronecker coefficients, which we call additive stability. Its main ingredient is the property of a matrix of being additive. This notion seems to be an important one: it appears in Discrete Tomography as a sufficient condition to uniqueness; it also appears in Manivel’s study of asymptotic properties of plethysm through Borel-Weil theory. The proof sketched here combines several results of the author on integer matrices motivated by Discrete Tomography with a new idea of Stembridge, that permits to bound some sequences of Kronecker coefficients. The advantage of additivity with respect to the previous approach by Stembridge is that it is very easy to produce new examples of additive matrices and, therefore, to produce many new examples of stability of Kronecker coefficients. We also show that Murnaghan’s stability property and other instances of stability discovered previously by the author are special cases of additive stability. Besides, our approach permits us to disprove a recent conjecture of Stembridge and to give a new characterization of additivity