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Abstract A minimal presentation of the cohomology ring of the flag manifold was given in A. Borel (1953). This presentation was extended by E. Akyildiz–A. Lascoux–P. Pragacz (1992) to a nonminimal one for all Schubert varieties. Work of V. Gasharov–V. Reiner (2002) gave a short, that is, polynomial‐size, presentation for a subclass of Schubert varieties that includes the smooth ones. In V. Reiner–A. Woo–A. Yong (2011), a general shortening was found; it implies an exponential upper bound of on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of on the number of generators needed in worst case, giving the first obstructions to short presentations. We prove the minimality conjecture. Our proof uses the Hopf algebra structure of the ring of symmetric functions.
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Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald famously expressing the principal specialization of any Schubert polynomial in terms of reduced words. We prove a lower bound on the principal specialization of dual characters of flagged Weyl modules. Our result yields an alternative proof of a conjecture of Stanley about the principal specialization of Schubert polynomials, originally proved by Weigandt.more » « less
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We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches ofarbitraryGrassmannian Schubert varieties.more » « less
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We give a new operator formula for Grothendieck polynomials that generalizes Magyar’s Demazure operator formula for Schubert polynomials. Our proofs are purely combinatorial, contrasting with the geometric and representation theoretic tools used by Magyar. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial.more » « less
