Related papers: The Wronski map and shifted tableau theory
We present a proof of a Littlewood-Richardson rule for the K-theory of odd orthogonal Grassmannians OG(n,2n+1), as conjectured in [Thomas-Yong '09]. Specifically, we prove that rectification using the jeu de taquin for increasing shifted…
We prove an analogue of the Mukhin-Tarasov-Varchenko theorem (formerly the Shapiro-Shapiro conjecture) for the maximal type B_n orthogonal Grassmannian OG(n,2n+1).
We give a embedding of the Lagrangian Grassmannian LG(n) inside an ordinary Grassmannian that is well-behaved with respect to the Wronski map. As a consequence, we obtain an analogue of the Mukhin-Tarasov-Varchenko theorem for LG(n). The…
We prove a generalization of the Shapiro-Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map,…
The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of…
The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite…
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(\lambda_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags…
Thurston obtained a combinatorial characterization for generic branched self-coverings that preserve the orientation of the oriented 2-sphere by associating a planar graph to them [arXiv:1502.04760]. In this work, the Thurston result is…
We give a new Littlewood-Richardson rule for the Schubert structure coefficients of isotropic Grassmannians, equivalently for the multiplication of $P$-Schur functions. Serrano (2010) previously gave a formula in terms of classes in his…
The Wronski map is a finite, PGL_2(C)-equivariant morphism from the Grassmannian Gr(d,n) to a projective space (the projectivization of a vector space of polynomials). We consider the following problem. If C_r < PGL_2(C) is a cyclic…
We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian…
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…
Given a $d$-dimensional vector space $V \subset \mathbb{C}[u]$ of polynomials, its Wronskian is the polynomial $(u + z_1) \cdots (u + z_n)$ whose zeros $-z_i$ are the points of $\mathbb{C}$ such that $V$ contains a nonzero polynomial with a…
We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of [Sch\"{u}tzenberger '77]'s theory of jeu de taquin. These are applied to the equivariant Schubert calculus of Grassmannians. Reinterpreting, we…
We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Sch\"{u}tzenberger '77] for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of…
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as…
We explain how genomic tableaux [Pechenik-Yong '15] are a semistandard complement to increasing tableaux [Thomas-Yong '09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth…
We give a proof of Pixton's generalized Faber-Zagier relations in the tautological Chow ring of $\overline M_{g,n}$. The strategy is very similar to the work of Pandharipande-Pixton-Zvonkine, who have given a proof of the same result in…
We address a unification of the Schubert calculus problems solved by [A. Buch '02] and [A. Knutson-T. Tao '03]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant K-theory of Grassmannians with…
We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case…