In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category Db(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GLk. This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection , which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of GLn−k. The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on Gr(k,n). We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z, and we identify the objects of Db(Gr(k,n)), which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in , by different methods.
The goal of the present paper is to extend the mitosis algorithm, originally developed by Ezra Miller and Allen Knutson for the case of Schubert polynomials, to the case of Grothendieck polynomials. In addition we will also use this algorithm to construct a short combinatorial proof of Fomin–Kirillov's formula for the coefficients of Grothendieck polynomials.
We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an SL(3, Z) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for N = 1 SQCD and N = 4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian N = 2 SCFT with E6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the N = 1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled N = 1 theory with enhanced E7 flavour symmetry.