In the late 1930's 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.

We provide counter-examples to Mulmuley's strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynmial time. We also provide a short proof of the #P-hardness of computing the Kronecker coefficients and another family of structural constants in the representation theory of the symmetric groups, Murnaghan's reduced Kronecker coefficients. An appendix by Mulmuley introduces a relaxed form of the saturation hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.

The volume function introduced by P. Milne to count the real roots of a system of polynomial equations in a box is a vector symmetric polynomial. Its expansion in the basis of monomial functions is provided. Only monomial functions of a particular kind appear in this expansion ("squarefree monomial functions"). By means of an appropriate specialization of the vector symmetric Newton identities, we derive an inductive formula to decompose the squarefree monomial functions in power sums. This formula is related to the lattice of the sub-hypergraphs of an hypergraph. As a corollary, an inductive formula to express the components of Milne's volume function in the power sums is obtained.

Using the a noncommutative version of Chevalley's theorem due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius series for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables.

The paper on arxiv ( on the left) is an expanded version of the published paper.

Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well-adapted to the study of the relative position of two conics defined by equations depending on parameters.

A matrix function, depending on a parameter t, and interpolating between the determinant and the permanent, is introduced. It is shown this function admits a simple expansion in terms of determinants and permanents of sub-matrices. This expansion is used to explain some formulas ocurring in the resolution of some systems of algebraic equations.

We investigate the normal forms of pure entangled three-partite qutrit states under local filtering operations. This amounts to study the orbits of the group $SL(3,C)^{\times 3}$ acting on $(C^3)^{\otimes 3}$. We describe the moduli space of this action 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 in a completely explicit way. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes.

Multisymmetric polynomials are the diagonal invariants of the symmetric group. Elementary multisymmetric polynomials are analogues of the elementary symmetric polynomials, in the multisymmetric setting. In this paper, we give a necessary and sufficient condition on a ring A for an algebra of multisymmetric polynomials with coefficients in A to be generated by the elementary multisymmetric polynomials.

We obtain a complete and minimal set of 170 generators for the algebra of $SL(2,C)^{\times 4}$-covariants of a four qubit system. This sheds light on the structure of the SLOCC orbits in the Hilbert space of this system, as well as on the more complicated algebra of its local unitary invariants.

This article is devoted to present, first, a family of formulas extending to the multivariate case the classical Newton Identities relating the coefficients of an univariate polynomial equation with its roots through the Power Sums and, secondly, the Generating Functions associated to the introduced Power Sums of the multivariate case. As a by-product the kinds of systems accepting these Newton Identities are also characterized together with those allowing the Power Sums to be computed in an inductive way directly from the coefficients of the polynomial system under consideration.

We consider the problem of providing systems of equations characterizing the forms with complex coefficients that are totally decomposable, i.e. products of linear forms. Our focus is computational. We present the well-known solution given at the end of the nineteenth century by Brill and Gordan and give a complete proof that their system does vanish only on the decomposable forms. We explore an idea due to Federico Gaeta which leads to an alternative system of equations, vanishing on the totally decomposable forms and on the forms admitting a multiple factor. Last, we give some insight on how to compute efficiently these systems of equations and point out possible further improvements.

A Steiner surface is the generic case of a quadratically parameterizable quartic surface used in geometric modeling. This paper studies quadratic parameterizations of surfaces under the angle of Classical Invariant Theory. Precisely, it exhibits a collection of covariants associated to projective quadratic parameterizations of surfaces, under the actions of linear reparameterization and linear transformations of the goal space. Each of these covariants comes with a simple geometric interpretation.

As an application, some of these covariants are used to produce explicit equations, inequations and inequalities defining the orbits of projective quadratic parameterizations of quartic surfaces.

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n large enough, 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 when this stable expansion is reached. We also compute two new bounds for the stabilization of a particular coefficient of such a product. Given partitions α and β, we give bounds for all the parts of any partition γ such that the corresponding Kronecker coefficient is nonzero. Finally, we also show that the reduced Kronecker coefficients are structure coefficients for the Heisenberg product introduced by Aguiar, Ferrer and Moreira.

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.

We provide a formula that recovers the Kronecker coefficients (the multiplicities of the irreducible representations in the tensor products of two irreducible representations of the symmetric group) from the reduced Kronecker coefficients (limits of certain stationary sequences of Kronecker coefficients introduced by Murnaghan). This formula generalizes a formula due to Rosas for Kronecker coefficients indexed by two two-row shapes. We use our formula to obtain a new stability bound for the Kronecker coefficients, and to describe explicitely the Kronecker coefficients indexed by two two-row shapes as a piecewise quasi-polynomial, with the chambers of a fan as domains of quasi-polynomiality.

We present two situations in which the diagonally symmetric polynomials of the n roots of a system of polynomial equations depend in a simple way on the coefficients of the equations: for Gröbner bases with prescribed leading terms, and for zero-dimensional strict complete intersections. The results are independent on the characteristic of the ground field.

The products of n linear forms in N variables are a subvariety of the space of the forms of degree n in N variables. At the end of the nineteenth century, Brill and Gordan used invariant theory to design a method to derive a system of equations defining set-theoretically this subvariety (Brill's equations). We show how to compute efficiently Brill's equations, and compare them with the ideal of the subvariety.

This is a translation, form german to french, of Paul Gordan's 1894 paper Das Zerfallen der Curven in gerade Linien (About the curves that decompose into straight lines).

Consider the couples of distinct proper non-empty real projective conics. A rigid isotopy for such an object is a continuous deformation of the defining equations of the two conics, not modifying the singularity of the intersection points. Each class of rigid isotopy corresponds to a ``configuration'' of a couple of conics. In the present paper, we show that if two couples of conics are in the same configuration, so are the corresponding couples of tangential conics. We make explicit the induced bijection between the configurations in the primal space and the configurations in the dual space.