Leonardo C. Mihalcea

Virginia Tech

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On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms

Duke Mathematical Journal (2007)

Leonardo C. Mihalcea

We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X=G/P. Using this formula, we give an effective algorithm to compute the 3-point, genus zero, equivariant Gromov-Witten invariants on X, which are the structure constants of its equivariant quantum cohomology algebra

Equivariant map

Zero (linguistics)

10.1215/s0012-7094-07-14024-9

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Citations (43)

Rational connectedness implies finiteness of quantum K-theory

arXiv (Cornell University) (2013)

Anders Skovsted Buch Pierre–Emmanuel Chaput Leonardo C. Mihalcea Nicolas Perrin

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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Rank (graph theory)

Flag (linear algebra)

Degree (music)

K-Theory

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Citations (3)

A conjectural Peterson isomorphism in K-theory

Journal of Algebra (2018)

Thomas Lam Changzheng Li Leonardo C. Mihalcea Mark Shimozono

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Isomorphism (crystallography)

Structure constants

Homology

Pontryagin's minimum principle

Affine variety

Isomorphism extension theorem

10.1016/j.jalgebra.2018.07.029

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Citations (21)

Curve neighborhoods of Schubert varieties

arXiv (Cornell University) (2013)

Anders Skovsted Buch Leonardo C. Mihalcea

A previous result of the authors with Chaput and Perrin states that the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space G/P is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety.

Schubert variety

Schubert Calculus

Equivariant map

Schubert polynomial

Degree (music)

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Citations (5)

Positivity of Peterson Schubert calculus

Advances in Mathematics (2024)

Rebecca Goldin Leonardo C. Mihalcea Rahul Kumar Singh

Schubert Calculus

Schubert polynomial

Schubert variety

10.1016/j.aim.2024.109879

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Citations (1)

Mather classes and conormal spaces of Schubert varieties in cominuscule spaces

Algebraic geometry (2023)

Leonardo C. Mihalcea Rahul Kumar Singh

Let G/P be a complex cominuscule flag manifold.We prove a type-independent formula for the torus-equivariant Mather class of a Schubert variety in G/P and for a Schubert variety pulled back via the natural projection G/Q → G/P for a parabolic subgroup Q ⊂ P .We apply this to find formulae for the local Euler obstructions of Schubert varieties and for the torus-equivariant localizations of the conormal spaces of these Schubert varieties.We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes, and we prove this conjecture in Lie types A, B, and D. We also conjecture that certain 'Mather polynomials' are unimodal in general Lie type and log concave in type A.

10.14231/ag-2023-019

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Citations (8)

Sums of binomial determinants, non-intersecting lattice paths and positivity of Chern-Schwartz-MacPherson classes

arXiv (Cornell University) (2007)

Leonardo C. Mihalcea

We give a combinatorial interpretation of a certain positivity conjecture of Chern-Schwartz-MacPherson classes, as stated by P. Aluffi and the author in a previous paper. It translates into a positivity property for a sum of p by p determinants consisting of binomial coefficients, generalizing the classical Theorem of Lindstrom-Gessel-Viennot et al. which computes these determinants in terms of non-intersecting lattice paths. We prove this conjecture for p=2,3.

Lattice (music)

Binomial coefficient

Binomial (polynomial)

10.48550/arxiv.math/0702566

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Citations (5)

Conjecture O holds for the odd symplectic Grassmannian

Bulletin of the London Mathematical Society (2019)

Changzheng Li Leonardo C. Mihalcea Ryan M. Shifler

Let $\mathrm{IG}(k, 2n+1)$ be the odd-symplectic Grassmannian. Property $\mathcal{O}$, introduced by Galkin, Golyshev and Iritani for arbitrary complex, Fano manifolds $X$, is a statement about the eigenvalues of the linear operator obtained by the quantum multiplication by the anticanonical class of $X$. We prove that property $\mathcal{O}$ holds in the case when $X= \mathrm{IG}(k, 2n+1)$ is an odd-symplectic Grassmannian. The proof uses the combinatorics of the recently found quantum Chevalley formula for $\mathrm{IG}(k, 2n+1)$, together with the Perron-Frobenius theory of nonnegative matrices.

Fano plane

Operator (biology)

10.1112/blms.12268

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Citations (12)

Equivariant Quantum Schubert Calculus

arXiv (Cornell University) (2004)

Leonardo C. Mihalcea

We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.

Equivariant map

Schubert Calculus

10.48550/arxiv.math/0406066

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Citations (0)

Positivity in equivariant quantum Schubert Calculus

arXiv (Cornell University) (2004)

Leonardo C. Mihalcea

We prove a positivity result in (T-)equivariant quantum cohomology of the homogeneous space G/P, generalizing Graham's positivity in equivariant cohomology.

Schubert Calculus

Equivariant map

10.48550/arxiv.math/0407239

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Citations (0)