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The theory of the exponential differential equations of semiabelian varieties
Selecta Mathematica - Tập 15 Số 3 - Trang 445-486 - 2009
Kirby, Jonathan
The complete first-order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arise from an amalgamation-with-predimension construction in the style of Hrushovski. The theories include necessary and sufficient conditions for a system of equations to have a solution. The necessary conditions generalize Ax’s differential fields v... hiện toàn bộ
On the toric ideals of matroids of a fixed rank
Selecta Mathematica - Tập 27 - Trang 1-17 - 2021
Michał Lasoń
In 1980 White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White’s conjecture for high degrees with respect to the rank. This extends our result (Lasoń and Michałek in Adv Math 259:1–12, 2014) confirming White’s conjecture ‘up to saturation’. Furthermore, we study degrees of Gröbner bases and Betti... hiện toàn bộ
Computational aspects of Calogero–Moser spaces
Selecta Mathematica - Tập 29 - Trang 1-46 - 2023
Cédric Bonnafé, Ulrich Thiel
We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the $$\mathbb {C}^\times $$ -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalizatio... hiện toàn bộ
Geometricity for derived categories of algebraic stacks
Selecta Mathematica - Tập 22 - Trang 2535-2568 - 2016
Daniel Bergh, Valery A. Lunts, Olaf M. Schnürer
We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smoo... hiện toàn bộ
Deformation cohomology of Schur–Weyl categories
Selecta Mathematica - Tập 29 - Trang 1-40 - 2022
Alexei Davydov, Mohamed Elbehiry
The deformation cohomology of a tensor category controls deformations of the constraint of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur–Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free... hiện toàn bộ
Constructible sheaves on nilpotent cones in rather good characteristic
Selecta Mathematica - Tập 23 - Trang 203-243 - 2016
Pramod N. Achar, Anthony Henderson, Daniel Juteau, Simon Riche
We study some aspects of modular generalized Springer theory for a complex reductive group G with coefficients in a field $$\Bbbk $$ under the assumption that the characteristic $$\ell $$ of $$\Bbbk $$ is rather good for G, i.e. $$\ell $$ is good and does not divide the order of the component group of the centre of G. We prove a comparison theorem relating the characteristic- $$\ell $$ generalized... hiện toàn bộ
t-Structures with Grothendieck hearts via functor categories
Selecta Mathematica - Tập 29 - Trang 1-73 - 2023
Manuel Saorín, Jan Št’ovíček
We study when the heart of a t-structure in a triangulated category $$\mathcal {D}$$ with coproducts is AB5 or a Grothendieck category. If $$\mathcal {D}$$ satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If $$\mathcal {D}$$ ... hiện toàn bộ
Remarks on eigenvalues and eigenvectors of Hermitian matrices, berry phase, adiabatic connections and quantum Hall effect
Selecta Mathematica - Tập 1 - Trang 1-19 - 1995
V. I. Arnold
The singularities of the stratification of the space of the Hermitian matrices according to the multiplicities of the eigenvalues are described as an informal complexification of the previous study of the space of the real symmetric matrices. The degeneration of the spectral sequence associated to this stratification provides some strange combinatorial identities. The eigenvector bundles over the ... hiện toàn bộ
Parameters and duality for the metaplectic geometric Langlands theory
Selecta Mathematica - Tập 24 - Trang 227-301 - 2017
D. Gaitsgory, S. Lysenko
We introduce the space of parameters for the metaplectic Langlands theory as factorization gerbes on the affine Grassmannian, and develop metaplectic Langlands duality in the incarnation of the metaplectic geometric Satake functor. We formulate a conjecture in the context of the global metaplectic Langlands theory, which is a metaplectic version of the “vanishing theorem” of Gaitsgory (Astérisque ... hiện toàn bộ
Local Langlands correspondence for even orthogonal groups via theta lifts
Selecta Mathematica - Tập 27 - Trang 1-71 - 2021
Rui Chen, Jialiang Zou
Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover, we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by ... hiện toàn bộ
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