### Working paper

## On the bounded derived category of $\mathsf{IGr}(3, 7)$

We introduce the notions of consistent pairs and consistent chains of $ t$-structures and prove that two consistent chains of $ t$-structures generate a distributive lattice. The technique developed is then applied to the pairs of chains obtained from the standard $ t$-structure on the derived category of coherent sheaves and the dual $ t$-structure by means of the shift functor. This yields a family of $ t$-structures whose hearts are known as perverse coherent sheaves. Access this article Login options Individual login Institutional loginvia Athens/Shibboleth The computer you are using is not registered by an institution with a subscription to this article. Please log in below. Find out more about journal subscriptions at your site. Purchase this article online Buy this article £33.00 (£39.60 incl. VAT) $59.70 US Dollar price There are no additional delivery charges. By purchasing this article, you are accepting IOP's Terms and Conditions for Document Delivery. If you would like to buy this article, but not online, please contact custserv@iop.org. Make a recommendation To gain access to this content, please complete the Recommendation Form and we will follow up with your librarian or Institution on your behalf. For corporate researchers we can also follow up directly with your R&D manager, or the information management contact at your company. Recommend this journal Institutional subscribers have access to the current volume, plus a 10-year back file (where available). Subscribe to this journal The title that you are trying to access is not part of the IOP Historic Archive. You can purchase a copy of the article that you wish to view. Related Articles Hyperplane sections and derived categories Derived categories of coherent sheaves and equivalences between them PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND DERIVED CATEGORIES OF COHERENT SHEAVES More Related Review Articles Classification of isomonodromy problems on elliptic curves Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field Geometric structures on moment-angle manifolds More

We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of k-dimensional subspaces in a vector space of dimension n. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when n and k are coprime these decompositions coincide and are minimal. In general, we conjecture minimality of the second decomposition.

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 [13], 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 [7], by different methods.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.