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The size of an arithmetical formula is the number of symbols (+,×) which it contains. The complexity of a polynomial defined over a field k is the minimum size of formulas defining it (see [10]). Using this notion of complexity, Valiant gave algebraic analogs to algorithmic complexity problems such as P = NP (see [10, 11, 12]). In this context, we would… (More)

Let G be a connected reductive subgroup of a complex connected reductive group Ĝ. Fix maximal tori and Borel subgroups of G and Ĝ. Consider the cone LR(G, Ĝ) generated by the pairs (ν, ν̂) of dominant characters such that Vν is a submodule of Vν̂ (with usual notation). Here we give a minimal set of inequalities describing LR(G, Ĝ) as a part of the dominant… (More)

- J. M. Landsberg, Laurent Manivel, Nicolas Ressayre
- ArXiv
- 2010

We determine set-theoretic defining equations for the variety Dualk,d,N ⊂ P(S d C N ) of hypersurfaces of degree d in C that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety GLn2 · [detn] ⊂ P(S n C n2), showing it is an irreducible component of the variety of hypersurfaces of degree n in C 2 with dual of… (More)

- N. Ressayre
- 2008

X(L) = {x ∈ X(L) such that Gx is finite and G · x is closed in X (L)} , where Gx is the stabilizer of x. It turns out that π −1 (π(x)) = G · x, for all x ∈ X(L). We refer to [7] or [8] for the classical properties of this quotient. Observe that this GIT-quotient is not canonical: it depends on a choice of an ample G-linearized line bundle L over X… (More)

- N. Ressayre
- 2008

Consider a partial flag variety X which is not a grassmaninan. Consider also its cohomology ring H∗(X,Z) endowed with the base formed by the Poincaré dual classes of the Schubert varieties. In [Ricar], E. Richmond showed that some coefficient structure of the product in H∗(X,Z) are products of two such coefficients for smaller flag varieties. Consider now a… (More)

- J. M. Landsberg, Nicolas Ressayre
- ITCS
- 2016

Grenet's determinantal representation for the permanent is optimal among determinantal representations that are equivariant with respect to left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal representation of the permanent must be polynomially related to… (More)

We give a new geometric proof of a conjecture of Fulton on the Littlewood-Richardson coefficients. This conjecture was firstly proved by Knutson, Tao and Woodward using the Honeycomb theory. A geometric proof was given by Belkale. Our proof is based on the geometry of Horn’s cones.

- N. Ressayre
- 2002

is symmetric around the origin. A plane configuration is said to be uniform if every pair of vectors is linearly independent. E. Cattani, A. Dickenstein and B. Sturmfels introduced this notion in [CDS99, CD02] for its relationship with multivariable hypergeometric functions in the sense of Gel’fand, Kapranov and Zelevinsky (see [GKZ89, GKZ90]). Balanced… (More)

- N Ressayre
- 2009

Let G be a complex connected reductive algebraic group and G/B denote the flag variety of G. A G-homogeneous space G/H is said to be spherical if H has a finite number of orbits in G/B. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group G (viewed as a G×G-homogeneous space) has particularly nice… (More)

In [3], Belkale and Kumar define a new product on the cohomology of flag varieties and use this new product to give an improved solution to the eigencone problem for complex reductive groups. In this paper, we give a generalization of the Belkale-Kumar product to the branching Schubert calculus setting. The study of Branching Schubert calculus attempts to… (More)