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We describe a relationship between work of Laksov, Gatto, and their collaborators on realizations of (generalized) Schubert calculus of Grassmannians, and the geometric Satake correspondence of Lusztig, Ginzburg, and Mirkovic and Vilonen. Along the way we obtain new proofs of equivariant Giambelli formulas for the ordinary and orthogonal Grassmannians, as well as a simple derivation of the rim-hook rule for computing in the equivariant quantum cohomology of the Grassmannian.
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We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we prove that the classes of the Kazhdan-Lusztig basis coincide with the classes determined by Zelevinskys small resolutions. These properties of the so-called KL-Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.
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.
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.
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.
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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