BRIAND , Emmanuel

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...

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 occurring in the resolution of some systems of algebraic equations.

Multisymmetric polynomials are the $r$-fold diagonal invariants of the symmetric group ${\mathfrak{S}}_n$. 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 the algebra of multisymmetric...

No description

We give three proofs of the following result conjectured by Carriegos, De Castro-García and Muñoz Castañeda in their work on enumeration of control systems: when ( k+1 2 ) ≤ n < ( k+2 2 ) , there are as many partitions of n with k corners as pairs of partitions (α, β) such that ( k+1 2 ) + |α| + |β| = n.

We give three proofs of the following result conjectured by Carriegos, De Castro-Garc\'ıa and Muñoz Castañeda in their work on enumeration of control systems: when (k+12)≤n<(k+22), there are as many partitions of n with k corners as pairs of partitions (α,β) such that (k+12)+|α|+|β|=n.

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...

Plethysm coefficients are important structural constants in the representation the- ory 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...

We compute with SageMath the group of all linear symmetries for the Littlewood-Richardson associated to the representations of SL3. We find that there are 144 symmetries, more than the 12 symmetries known for the Littlewood-Richardson coefficients in general.

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 Milne's volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular...

No description

Ministerio de Economía y Competitividad MTM2016-75024-P

We study the rate of growth experienced by the Kronecker coefficients as we add cells to the rows and columns indexing partitions. We do this by moving to the setting of the reduced Kronecker coefficients.

We obtain a complete and minimal set of 170 generators for the algebra of SL(2, C )×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...

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...

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...

This text is an appendix to our work "On the growth of Kronecker coefficients" [1]. Here, we provide some complementary theorems, re- marks, and calculations that for the sake of space are not going to appear into the final version of our paper. We follow the same terminology and notation. External references to numbered equations, theorems,...

We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coe cients, Kronecker coe cients, plethysm coe cients, and the Kostka{Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.

The SU(3) tensor multiplicities are piecewise polynomial of degree 1 in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order 144) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as...

We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.

Nous montrons que plusieurs des principales constantes de structure de la theorie des fonctions symétriques (les coefficients de Littlewood-Richardson, les coefficients de Kronecker, les coefficients du plethysme, et les polynômes de Kostka-Foulkes) ont en commun des symetries décrites par des opérations de complémentation dans des rectangles...

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...

We investigate the combinatorics of the general formulas for the powers of the operator h∂k, where h is a central element of a ring and ∂ is a differential operator. This generalizes previous work on the powers of operators h∂. New formulas for the generalized Stirling numbers are obtained.

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...