Emmanuel Briand

A complete set of covariants of the four qubit system

We obtain a complete and minimal set of 170 generators for the algebra of SL(2, )×4 covariants of a binary quadrilinear form. Interpreted in terms of a four qubit system, this describes in particular the algebraic varieties formed by the orbits of local filtering operations in its projective Hilbert space. Also, this sheds some light on the local unitary invariants, and provides all the possible building blocks for the construction of entanglement measures for such a system.

The moduli space of three-qutrit states

We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behavior 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 parametrized 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.

Reduced Kronecker Coefficients and Counter–Examples to Mulmuley’s Strong Saturation Conjecture SH: With an Appendix by Ketan Mulmuley

Abstract.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 polynomial time. We also provide a short proof of the #P– hardness of computing the Kronecker coefficients. Both results rely on the connections between 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.

Milne's volume function and vector symmetric polynomials

The number of real roots of a system of polynomial equations fitting inside a given box can be counted using a vector symmetric polynomial introduced by P. Milne, the volume function. We provide the expansion of Milnes volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear in the expansion, the squarefree monomial functions. By means of an appropriate specialization of the vector symmetric Newton identities, we derive an inductive formula that expresses the squarefree monomial functions in the power sums basis. As a corollary, we obtain an inductive formula that writes Milnes volume function in the power sums basis. The lattice of the sub-hypergraphs of a hypergraph appears in a natural way in this setting.

Covariants Vanishing on Totally Decomposable Forms

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.

MULTIVARIATE NEWTON SUMS: IDENTITIES AND GENERATING FUNCTIONS

ABSTRACT This paper is devoted to present, first, a family of formulas extending to the multivariate case the classical Newton (or Newton–Girard) Identities relating the coefficients of a univariate polynomial equation with its roots through the Newton Sums and, secondly, the Generating Functions associated to the new introduced Newton 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 Newton Sums to be computed in an inductive way directly from the coefficients of the polynomial system under consideration.

Equations, inequations and inequalities characterizing the configurations of two real projective conics

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.

Some Covariants Related to Steiner Surfaces

ABSTRACT. 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 target space. Each of these covariants comes with a simple geometric interpretation. As an application, some of these covariants are used to produce explicit equations and inequalities defining the orbits of projective quadratic parameterizations of quartic surfaces.

On the Sn-module structure of the noncommutative harmonics

Using a noncommutative analog of Chevalleys decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius characteristic 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.

Combinatorial proof for a stability property of plethysm coefficients

Plethysm coefficients are important structural constants in the representation theory of the symmetric groups and general linear groups. Remarkably, some sequences of plethysm coefficients stabilize (they are ultimately constants). In this paper we give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose plethysm coefficients as a alternating sum of terms counting integer points in polytopes, and exhibit bijections between these sets of integer points.