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Register a new userSchubert calculus and Gelfand-Zetlin polytopes
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We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.
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I construct a correspondence between the Schubert cycles on the variety of complete flags in C^n and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SL_n(C) with a strictly dominant highest weight. The construction is based on a geometric presentation of Schubert cells by Bernstein-Gelfand-Gelfand using Demazure modules. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand-Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.
A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the general framework and survey particular realizations of this approach in types A, B and C.
In the recent paper [arXiv:1612.06893] P. Burgisser and A. Lerario introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $delta_{k,n}$ the average number of projective $k$-planes in $mathbb{R}textrm{P}^n$ that intersect $(k+1)(n-k)$ many random, independent and uniformly distributed linear projective subspaces of dimension $n-k-1$. They called $delta_{k,n}$ the expected degree of the real Grassmannian $mathbb{G}(k,n)$ and, in the case $k=1$, they proved that: $$ delta_{1,n}= frac{8}{3pi^{5/2}} cdot left(frac{pi^2}{4}right)^n cdot n^{-1/2} left( 1+mathcal{O}left(n^{-1}right)right) .$$ Here we generalize this result and prove that for every fixed integer $k>0$ and as $nto infty$, we have begin{equation*} delta_{k,n}=a_k cdot left(b_kright)^ncdot n^{-frac{k(k+1)}{4}}left(1+mathcal{O}(n^{-1})right) end{equation*} where $a_k$ and $b_k$ are some (explicit) constants, and $a_k$ involves an interesting integral over the space of polynomials that have all real roots. For instance: $$delta_{2,n}= frac{9sqrt{3}}{2048sqrt{2pi}} cdot 8^n cdot n^{-3/2} left( 1+mathcal{O}left(n^{-1}right)right).$$ Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and we give an explicit formula for $delta_{1,n}$ involving a one dimensional integral of certain combination of Elliptic functions.
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation. A similarly rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation. This showcases new possibilities for the use of computers in mathematical research.
The Macaulay2 package NumericalSchubertCalculus provides methods for the numerical computation of Schubert problems on Grassmannians. It implements both the Pieri homotopy algorithm and the Littlewood-Richardson homotopy algorithm. Each algorithm has two independent implementations in this package. One is in the scripting language of Macaulay2 using the package NumericalAlgebraicGeometry, and the other is in the compiled code of PHCpack.
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