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Pierre-Emmanuel Chaput

Publications and source records attributed to Pierre-Emmanuel Chaput.

At least 19 recordsLinked to original sources
arXiv2024

Equivariant rigidity of Richardson varieties

We prove that Schubert and Richardson varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under suitable conditions. This is used to prove that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is a Schubert variety. We pose a stronger conjecture which implies a Seidel multiplication formula in equivariant quantum K-theory, and prove this conjecture for cominuscule flag varieties.

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arXiv2023

Seidel and Pieri products in cominuscule quantum K-theory

We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space.

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arXiv2022

Bidilatation of Small Littlewood-Richardson Coefficients

The Littlewood-Richardson coefficients $c^ν_{λ,μ}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the triples of partitions $(λ, μ, ν)$ of length at most $n$. By the so-called Fulton conjecture, if $c^ν_{λ,μ}=1$ then $c^{kν}_{kλ,kμ}= 1$, for any $k \geq 0$. Similarly, as proved by Ikenmeyer or Sherman, if $c^ν_{λ,μ}=2$ then $c^{kν}_{kλ,kμ} = k + 1$, for any $k\geq 0$. Here, given a partition $λ$, we set $λ(p, q) = p(qλ')'$ , where prime denotes the conjugate partition. We observe that Fulton's conjecture implies that if $c^ν_{λ,μ}=1$ then $c^{ν(p,q)}_{λ(p,q),μ(p,q)}=1$, for any $p, q \geq 0$. Our main result is that if $c^ν_{λ,μ}=2$ then $c^{ν(p,q)}_{λ(p,q),μ(p,q)}$ is the binomial $\begin{pmatrix} p+q\\ q \end{pmatrix}$, for any $p, q \geq 0$.

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arXiv2022

Positivity of minuscule quantum K-theory

We prove that the Schubert structure constants of the quantum $K$-theory ring of any minuscule flag variety or quadric hypersurface have signs that alternate with codimension. We also prove that the powers of the deformation parameter $q$ that occur in the product of two Schubert classes in the quantum cohomology or quantum $K$-theory ring of a cominuscule flag variety form an integer interval. Our proofs are based on several new results, including an explicit description of the most general non-empty intersection of two Schubert varieties in an arbitrary flag manifold, and a computation of the cohomology groups of any negative line bundle restricted to a Richardson variety in a cominuscule flag variety. We also give a type-uniform proof of the quantum-to-classical theorem, which asserts that the (3-point, genus 0) Gromov-Witten invariants of any cominuscule flag variety are classical triple-intersection numbers on an associated flag variety. Finally, we prove several new results about the geometry and combinatorics related to this theorem.

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arXiv2020

Affine symmetries in quantum cohomology: corrections and new results

In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas.

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arXiv2020

Parametrization, structure and Bruhat order of certain spherical quotients

Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits and we show that each $Z$-orbit has a structure of algebraic affine bundle. In particular, in type $A$, we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type $A$, we show that the Bruhat order of the $Z$-orbits can be described in this way.

math.RT↗
arXiv2017

Projected Gromov-Witten varieties in cominuscule spaces

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G/P. When X is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3 point, genus zero) K-theoretic Gromov-Witten invariants of X are determined by the projected Gromov-Witten varieties, which extends an earlier result of Knutson, Lam, and Speyer. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety.

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arXiv2017

A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties

We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the equivariant quantum $K$-theory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum $K$-theory of Grassmannians of Lie type A.

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arXiv2017

Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $ρ$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506.07274). However we do not focus immediately on the exceptional cases and instead we provide a more unified perspective, as independent as possible of the classification of the simple Hermitian Lie groups. This relies on the study of the cominuscule representation of the complexification of the target group. As a by-product of our methods, when the target Hermitian group $G$ has tube type, we obtain an inequality on the Toledo invariant of the representation $ρ:Γ\rightarrow G$ which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).

math.DG↗
arXiv2015

Stability of the tangent bundle of G/P in positive characteristics

Let $G$ be an almost simple simply-connected affine algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. If $G$ has type $B_n$, $C_n$ or $F_4$, we assume that $p > 2$, and if $G$ has type $G_2$, we assume that $p > 3$. Let $P \subset G$ be a parabolic subgroup. We prove that the tangent bundle of $G/P$ is Frobenius stable with respect to the anticanonical polarization on $G/P$.

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arXiv2013

Rational connectedness implies finiteness of quantum K-theory

Let X be any generalized flag variety with Picard group of rank one. Given a degree d, consider the Gromov-Witten variety of rational curves of degree d in X that meet three general points. We prove that, if this Gromov-Witten variety is rationally connected for all large degrees d, then the structure constants of the small quantum K-theory ring of X vanish for large degrees.

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arXiv2012

Finiteness of cominuscule quantum K-theory

The product of two Schubert classes in the quantum K-theory ring of a homogeneous space X = G/P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when X is cominuscule, all boundary Gromov-Witten varieties defined by three single points in X are rationally connected.

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arXiv2012

On the equivariant cohomology of Hilbert schemes of points in the plane

Let $S$ be the affine plane regarded as a toric variety with an action of the 2-dimensional torus $T$. We study the equivariant Chow ring $A_{K}^*(Hilb^n(S))$ of the punctual Hilbert scheme $Hilb^n(S)$ with equivariant coefficients inverted. We compute base change formulas in $A_{K}^*(Hilb^n(S))$ between the natural bases introduced by Nakajima, Ellingsrud and Strømme, and the classical basis associated with the fixed points. We compute the equivariant commutation relations between creation/annihilation operators. We express the class of the small diagonal in $Hilb^n(S)$ in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme $Hilb^[n,n+1](S)$ parametrizing nested punctual subschemes of degree $n$ and $n+1$ is irreducible.

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arXiv2010

On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces

In this paper we consider homaloidal polynomial functions $f$ such that their multiplicative Legendre transform $f_*$, defined as in \cite[Section3.2]{MR1890194}, is again polynomial. Following Dolgachev \cite{MR1786486}, we call such polynomials EKP-homaloidal. We prove that every EKP-homaloidal polynomial function of degree three is a relative invariant of a symmetric prehomogeneous vector space. This provides a complete proof of \cite[Theorem 3.10, p.~39]{MR1890194}. With respect to the original argument of Etingof, Kazhdan and Polischuk our argument focuses more on prehomogeneous vector spaces and, in principle, it may suggest a way to attack the more general problem raised in \cite[Section 3.4]{MR1890194} of classification of EKP-homaloidal polynomials of arbitrary degree.

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arXiv2009

Rationality of some Gromov-Witten varieties and application to quantum K-theory

We show that for any minuscule or cominuscule homogeneous space X, the Gromov-Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of A. Buch and L. Mihalcea to constructions of the authors together with L. Manivel, we deduce that the equivariant K-theoretic three points Gromov-Witten invariants are equal to classical equivariant K-theoretic invariants on auxilliary spaces.

math.AG↗
arXiv2009

On the quantum cohomology of adjoint varieties

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown.

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