Kevin Purbhoo

University of Waterloo

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Jake Levinson

Université de Montréal

Frank Sottile

Texas A&M University

Maria Gillespie

Colorado State University

Harry Tamvakis

University of Maryland, College Park

Andrew Kresch

University of Zurich

Anders Skovsted Buch

Laboratoire de Mathématiques

Allen Knutson

Cornell University

Donguk Rhee

Research Studios Austria

Shelley Wu

University of Waterloo

Steven N. Karp

University of Notre Dame

Cooperative Institutions

University of Waterloo

University of Pennsylvania

University of Zurich

Texas A&M University

University of Washington

Colorado State University

University of Michigan–Ann Arbor

University of California, Berkeley

Cornell University

University of Maryland, College Park

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Shifted tableaux crystals

arXiv (Cornell University) (2017)

Maria Gillespie Jake Levinson Kevin Purbhoo

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur $Q$-functions. We give local axioms for these crystals, which closely resemble the Stembridge axioms for type A. Finally, we give a new criterion for such tableaux to be ballot.

10.48550/arxiv.1711.06919

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The Wronski map and shifted tableau theory

arXiv (Cornell University) (2010)

Kevin Purbhoo

The Mukhin-Tarasov-Varchenko Theorem, conjectured by B. and M. Shapiro, has a number of interesting consequences. Among them is a well-behaved correspondence between certain points on a Grassmannian - those sent by the Wronski map to polynomials with only real roots - and (dual equivalence classes of) Young tableaux. In this paper, we restrict this correspondence to the orthogonal Grassmannian OG(n,2n+1) inside Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and only if the corresponding tableau has a certain type of symmetry. From this we recover much of the theory of shifted tableaux for Schubert calculus on OG(n,2n+1), including a new, geometric proof of the Littlewood-Richardson rule for OG(n,2n+1).

Schubert Calculus

Young tableau

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Schubert curves in the orthogonal Grassmannian

arXiv (Cornell University) (2019)

Maria Gillespie Jake Levinson Kevin Purbhoo

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 osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy operator $\omega$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute $\omega$ without rectifying the skew tableau. The first uses recently-developed shifted tableau crystal operators, while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves.

Schubert Calculus

Osculating circle

Operator (biology)

10.48550/arxiv.1903.01673

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Compression of root systems and the E-sequence

arXiv (Cornell University) (2006)

Kevin Purbhoo

We examine certain maps from root systems to vector spaces over finite fields. By choosing appropriate bases, the images of these maps can turn out to have nice combinatorial properties, which reflect the structure of the underlying root system. The main examples are E_6 and E_7.

Root (linguistics)

Sequence (biology)

Root system

10.48550/arxiv.math/0609014

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The Wronski map and shifted tableau theory

arXiv (Cornell University) (2010)

Kevin Purbhoo

Schubert Calculus

Young tableau

10.48550/arxiv.1009.0035

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A vanishing and a non-vanishing condition for Schubert calculus on G/B

arXiv (Cornell University) (2003)

Kevin Purbhoo

Schubert Calculus

Schubert variety

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A topological proof of the Shapiro-Shapiro conjecture

arXiv (Cornell University) (2019)

Jake Levinson Kevin Purbhoo

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, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given as an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

Wronskian

Complex conjugate

Degree (music)

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Minimal orbits of promotion

arXiv (Cornell University) (2014)

Kevin Purbhoo Donguk Rhee

We give a bijection between the symmetric group S_n, and the set of standard Young tableaux of rectangular shape m^n, m<=n, that have order n under jeu de taquin promotion.

Promotion (chess)

Young tableau

10.48550/arxiv.1407.0078

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Class groups of open Richardson varieties in the Grassmannian are trivial

Journal of Commutative Algebra (2022)

Jake Levinson Kevin Purbhoo

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing the coordinate ring at a suitable set of Plücker coordinates. We prove that these Plücker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.

Affine variety

10.1216/jca.2022.14.267

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Product and puzzle formulae for GL_n Belkale-Kumar coefficients

arXiv (Cornell University) (2010)

Allen Knutson Kevin Purbhoo

The Belkale-Kumar product on H*(G/P) is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case G=GL_n, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for G/P a (d-1)-step flag manifold, each Belkale-Kumar structure constant is a product of d(d-1)/2 Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into d-1 factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward~'04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always 1; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension 1 or 2.

Flag (linear algebra)

Structure constants

Constant (computer programming)

Cohomology ring

Generalized flag variety

Cup product

10.48550/arxiv.1008.4979

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