pinecone/core-2020-05-10-deduplication huggingface dataset:
Gaudin subalgebras and wonderful models

The final publication is available at Springer via http://dx.doi.org/10.1007/s00029-015-0213-y.Gaudin Hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin Hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini–Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes
Tropical Theta Functions and Log Calabi-Yau Surfaces

We generalize the standard combinatorial techniques of toric geometry to the
study of log Calabi-Yau surfaces. The character and cocharacter lattices are
replaced by certain integral linear manifolds described by Gross, Hacking, and
Keel, and monomials on toric varieties are replaced with the canonical theta
functions which GHK defined using ideas from mirror symmetry. We describe the
tropicalizations of theta functions and use them to generalize the dual pairing
between the character and cocharacter lattices. We use this to describe
generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and
finite Fourier series expansions. We hope that these techniques will generalize
to higher-rank cluster varieties.Comment: 40 pages, 2 figures. The final publication is available at Springer
  via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016
Characteristic classes and Hilbert-Poincar\'e series for perverse
  sheaves on abelian varieties

The convolution powers of a perverse sheaf on an abelian variety define an
interesting family of branched local systems whose geometry is still poorly
understood. We show that the generating series for their generic rank is a
rational function of a very simple shape and that a similar result holds for
the symmetric convolution powers. We also give formulae for other Schur
functors in terms of characteristic classes on the dual abelian variety, and as
an example we discuss the case of Prym-Tjurin varieties.Comment: Some typos removed, references update
Constructible sheaves on nilpotent cones in rather good characteristic

We study some aspects of modular generalized Springer theory for a complex
reductive group $G$ with coefficients in a field $\mathbb k$ under the
assumption that the characteristic $\ell$ of $\mathbb k$ 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 Springer correspondence to the
characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$
`cleanness conjecture'; we prove it in some cases; and we deduce several
consequences, including a classification of supercuspidal sheaves and an
orthogonal decomposition of the equivariant derived category of the nilpotent
cone.Comment: 26 page
The polytope of Tesler matrices

We introduce the Tesler polytope Tes_n(a_1,a_2,...,a_n), whose integer points
are the Tesler matrices of size n with nonnegative integer hook sums
a_1,a_2,...,a_n. We show that Tes_n(a) is a flow polytope and therefore the
number of Tesler matrices is counted by the type A_n Kostant partition function
evaluated at (a_1,a_2,...,a_n,-a_1-...-a_n). We describe the faces of this
polytope in terms of "Tesler tableaux" and characterize when the polytope is
simple. We prove that the h-vector of Tes_n(a) when all a_i>0 is given by the
Mahonian numbers and calculate the volume of Tes_n(1,1,...,1) to be a product
of consecutive Catalan numbers multiplied by the number of standard Young
tableaux of staircase shape.Comment: 26 pages, 4 figures. v3 has a corrected proof of Lemma 2.4, updated
  reference
Multivariable $(\varphi,\Gamma)$-modules and smooth $o$-torsion
  representations

Let $G$ be a $\mathbb{Q}_p$-split reductive group with connected centre and
Borel subgroup $B=TN$. We construct a right exact functor $D^\vee_\Delta$ from
the category of smooth modulo $p^n$ representations of $B$ to the category of
projective limits of finitely generated \'etale $(\varphi,\Gamma)$-modules over
a multivariable (indexed by the set of simple roots) commutative Laurent-series
ring. These correspond to representations of a direct power of
$\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ via an equivalence of
categories. Parabolic induction from a subgroup $P=L_PN_P$ corresponds to a
basechange from a Laurent-series ring in those variables with corresponding
simple roots contained in the Levi component $L_P$. $D^\vee_\Delta$ is exact
and yields finitely generated objects on the category $SP_A$ of finite length
representations with subquotients of principal series as Jordan-H\"older
factors. Lifting the functor $D^\vee_\Delta$ to all (noncommuting) variables
indexed by the positive roots allows us to construct a $G$-equivariant sheaf
$\mathfrak{Y}_{\pi,\Delta}$ on $G/B$ and a $G$-equivariant continuous map from
the Pontryagin dual $\pi^\vee$ of a smooth representation $\pi$ of $G$ to the
global sections $\mathfrak{Y}_{\pi,\Delta}(G/B)$. We deduce that
$D^\vee_\Delta$ is fully faithful on the full subcategory of $SP_A$ with
Jordan-H\"older factors isomorphic to irreducible principal series.Comment: 55 pages, revised, to appear in Selecta Mathematic
Combined tilings and separated set-systems

In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated
collections of subsets of the ordered $n$-element set $[n]$ (using this notion
to give a combinatorial characterization for quasi-commuting minors of a
quantum matrix). They conjectured the purity of certain natural domains
$D\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the
hyper-simplex $\{X\subseteq[n]\colon |X|=m\}$ for $m$ fixed), where $D$ is
called pure if all maximal weakly separated collections in $D$ have the same
cardinality. These conjectures have been answered affirmatively.
  In this paper, generalizing those earlier results, we reveal wider classes of
pure domains in $2^{[n]}$. This is obtained as a consequence of our study of a
novel geometric--combinatorial model for weakly separated set-systems,
so-called \emph{combined (polygonal) tilings} on a zonogon, which yields a new
insight in the area.Comment: 30 pages. Revised version. To appear in Selecta Mathematic
Root systems, spectral curves, and analysis of a Chern-Simons matrix
  model for Seifert fibered spaces

We study a class of scalar, linear, non-local Riemann-Hilbert problems (RHP)
involving finite subgroups of PSL(2,C). We associate to such problems a (maybe
infinite) root system and describe the relevance of the orbits of the Weyl
group in the construction of its solutions. As an application, we study in
detail the large N expansion of SU(N) or SO(N) or Sp(2N) Chern-Simons partition
function Z_N(M) of 3-manifolds M that are either rational homology spheres or
more generally Seifert fibered spaces. It has a matrix model-like
representation, whose spectral curve can be characterized in terms of a RHP as
above. When pi_1(M) is finite (i.e. for manifolds M that are quotients of
\mathbb{S}_{3} by a finite isometry group of type ADE), the Weyl group
associated to the RHP is finite and the spectral curve is algebraic and can be
in principle computed. We then show that the large $N$ expansion of Z_N(M) is
computed by the topological recursion. This has consequences for the
analyticity properties of SU/SO/Sp perturbative invariants of knots along
fibers in $M$.Comment: 92 pages, 20 figures. Section 9 by Alexander Wei{\ss}
Geometricity for derived categories of algebraic stacks

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 smooth, projective variety. The same holds for a
smooth, projective, tame Artin stack over an arbitrary field.Comment: 31 page
The Schwartz space of a smooth semi-algebraic stack

Schwartz functions, or measures, are defined on any smooth semi-algebraic
("Nash") manifold, and are known to form a cosheaf for the semi-algebraic
restricted topology. We extend this definition to smooth semi-algebraic stacks,
which are defined as geometric stacks in the category of Nash manifolds.
  Moreover, when those are obtained from algebraic quotient stacks of the form
X/G, with X a smooth affine variety and G a reductive group defined over a
global field k, we define, whenever possible, an "evaluation map" at each
semisimple k-point of the stack, without using truncation methods. This
corresponds to a regularization of the sum of those orbital integrals whose
semisimple part corresponds to the chosen k-point.
  These evaluation maps produce, in principle, a distribution which generalizes
the Arthur-Selberg trace formula and Jacquet's relative trace formula, although
the former, and many instances of the latter, cannot actually be defined by the
purely geometric methods of this paper. In any case, the stack-theoretic point
of view provides an explanation for the pure inner forms that appear in many
versions of the Langlands, and relative Langlands, conjectures.Comment: 96pp. Erratum added at the end to fix two gaps and strengthen some
  statements from quasi-isomorphism to homotopy equivalenc
Towards a cluster structure on trigonometric zastava

We study a moduli problem on a nodal curve of arithmetic genus 1, whose
solution is an open subscheme in the zastava space for projective line. This
moduli space is equipped with a natural Poisson structure, and we compute it in
a natural coordinate system. We compare this Poisson structure with the
trigonometric Poisson structure on the transversal slices in an affine flag
variety. We conjecture that certain generalized minors give rise to a cluster
structure on the trigonometric zastava.Comment: Main text by M. Finkelberg, A. Kuznetsov and L. Rybnikov with an
  appendix by G. Dobrovolska; v3 32 pages, Proof of Proposition 4.3 corrected,
  Section 1.5 added; v4 33 pages, the final version to appear in Selecta Math.;
  v5 the published versio
Coherent rings, fp-injective modules, dualizing complexes, and covariant
  Serre-Grothendieck duality

For a left coherent ring A with every left ideal having a countable set of
generators, we show that the coderived category of left A-modules is compactly
generated by the bounded derived category of finitely presented left A-modules
(reproducing a particular case of a recent result of Stovicek with our
methods). Furthermore, we present the definition of a dualizing complex of
fp-injective modules over a pair of noncommutative coherent rings A and B, and
construct an equivalence between the coderived category of A-modules and the
contraderived category of B-modules. Finally, we define the notion of a
relative dualizing complex of bimodules for a pair of noncommutative ring
homomorphisms A \to R and B \to S, and obtain an equivalence between the
R/A-semicoderived category of R-modules and the S/B-semicontraderived category
of S-modules. For a homomorphism of commutative rings A\to R, we also construct
a tensor structure on the R/A-semicoderived category of R-modules. A vision of
semi-infinite algebraic geometry is discussed in the introduction.Comment: LaTeX 2e with pb-diagram and xy-pic, 30 pages, 2 commutative
  diagrams; v.3: several misprints corrected, expositional improvement in
  Section 2; v.4: examples added in the introduction and in Sections 3 and 5;
  v.5: more misprints corrected, new Section 6 added; v.6: exposition improved
  in Section 6 -- this is intended as the final versio
Rank 2 wall-crossing and the Serre correspondence

We study Quot schemes of 0-dimensional quotients of sheaves on 3-folds $X$.
When the sheaf $\mathcal{R}$ is rank 2 and reflexive, we prove that the
generating function of Euler characteristics of these Quot schemes is a power
of the MacMahon function times a polynomial. This polynomial is itself the
generating function of Euler characteristics of Quot schemes of a certain
0-dimensional sheaf, which is supported on the locus where $\mathcal{R}$ is not
locally free.
  In the case $X = \mathbb{C}^3$ and $\mathcal{R}$ is equivariant, we use our
result to prove an explicit product formula for the generating function. This
formula was first found using localization techniques in previous joint work
with B. Young. Our results follow from R. Hartshorne's Serre correspondence and
a rank 2 version of a Hall algebra calculation by J. Stoppa and R.P. Thomas.Comment: 20 pages. Published version. Addition to published version: the
  assumptions in Thm. 1.2 can be weakened due to an argument from J. Rennemo
  (Section 1.2
A cohomological obstruction to the existence of compact Clifford-Klein
  forms

In this paper, we continue the study of the existence problem of compact
Clifford-Klein forms from a cohomological point of view, which was initiated by
Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an
obstruction to the existence of compact Clifford-Klein forms by relating a
natural homomorphism from relative Lie algebra cohomology to de Rham cohomology
with an upper-bound estimate for cohomological dimensions of discontinuous
groups. From this obstruction, we derive some examples, e.g.
$\mathrm{SO}_0(p+r, q)/(\mathrm{SO}_0(p,q) \times \mathrm{SO}(r))$ $(p,q,r \geq
1, \ q:\text{odd})$ and $\mathrm{SL}(p+q, \mathbb{C})/\mathrm{SU}(p,q)$ $(p,q
\geq 1)$, of a homogeneous space that does not admit a compact Clifford-Klein
form. To construct these examples, we apply H. Cartan's theorem on relative Lie
algebra cohomology of reductive pairs and the theory of $\epsilon$-families of
semisimple symmetric pairs.Comment: 18 page
Shelah's eventual categoricity conjecture in universal classes. Part II

We prove that a universal class categorical in a high-enough cardinal is
categorical on a tail of cardinals. As opposed to other results in the
literature, we work in ZFC, do not require the categoricity cardinal to be a
successor, do not assume amalgamation, and do not use large cardinals. Moreover
we give an explicit bound on the "high-enough" threshold:
  $\mathbf{Theorem}$
  Let $\psi$ be a universal $\mathbb{L}_{\omega_1, \omega}$ sentence. If $\psi$
is categorical in some $\lambda \ge \beth_{\beth_{\omega_1}}$, then $\psi$ is
categorical in all $\lambda' \ge \beth_{\beth_{\omega_1}}$.
  As a byproduct of the proof, we show that a conjecture of Grossberg holds in
universal classes:
  $\mathbf{Corollary}$
  Let $\psi$ be a universal $\mathbb{L}_{\omega_1, \omega}$ sentence that is
categorical in some $\lambda \ge \beth_{\beth_{\omega_1}}$, then the class of
models of $\psi$ has the amalgamation property for models of size at least
$\beth_{\beth_{\omega_1}}$.
  We also establish generalizations of these two results to uncountable
languages. As part of the argument, we develop machinery to transfer
model-theoretic properties between two different classes satisfying a
compatibility condition. This is used as a bridge between Shelah's milestone
study of universal classes (which we use extensively) and a categoricity
transfer theorem of the author for abstract elementary classes that have
amalgamation, are tame, and have primes over sets of the form $M \cup \{a\}$.Comment: 49 page
Trace Identities for the Topological Vertex

The topological vertex is a universal series which can be regarded as an
object in combinatorics, representation theory, geometry, or physics. It
encodes the combinatorics of 3D partitions, the action of vertex operators on
Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the
open string partition function of $\mathbb{C}^3$.
  We prove several identities in which a sum over terms involving the
topological vertex is expressed as a closed formula, often a product of simple
terms, closely related to Fourier expansions of Jacobi forms. We use purely
combinatorial and representation theoretic methods to prove our formulas, but
we discuss applications to the Donaldson-Thomas invariants of elliptically
fibered Calabi-Yau threefolds at the end of the paper.Comment: 21 page
A class of knots with simple $SU(2)$ representations

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the
fundamental group of its complement which map a meridian to a trace-free
element in $SU(2)$ are binary dihedral. This is a generalisation of being a
2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not
$SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number
$\geq 3$ which are $SU(2)$-simple.
  One expects the instanton knot Floer homology $I^\natural(K)$ of a
$SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot
determinant $\det(K)$. In fact, the complex underlying $I^\natural(K)$ is of
rank equal to $\det(K)$, provided a genericity assumption holds that is
reasonable to expect. Thus formally there is a resemblance to strong L-spaces
in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we
introduce this formal resemblance is reflected topologically: The branched
double covers of these knots are strong L-spaces. In fact, somewhat
surprisingly, these knots are alternating. However, the Conway spheres are
hidden in any alternating diagram.
  With the methods we use, we show that an integer homology 3-sphere which is a
graph manifold always admits irreducible representations of its fundamental
group.Comment: 22 pages, 10 figures, to appear in Selecta Mathematic
Screening operators for W-algebras

Let $\mathfrak{g}$ be a simple finite-dimensional Lie superalgebra with a
non-degenerate supersymmetric even invariant bilinear form, $f$ a nilpotent
element in the even part of $\mathfrak{g}$, $\Gamma$ a good grading of
$\mathfrak{g}$ for $f$ and $\mathcal{W}^{k}(\mathfrak{g},f;\Gamma)$ the
$\mathcal{W}$-algebra associated with $\mathfrak{g},f,k,\Gamma$ defined by the
generalized Drinfeld-Sokolov reduction. In this paper, we present each
$\mathcal{W}$-algebra as the intersection of kernels of the screening
operators, acting on the tensor vertex superalgebra of an affine vertex
superalgebra and a neutral free superfermion vertex superalgebra. As
applications, we prove that the $\mathcal{W}$-algebra associated with a regular
nilpotent element in $\mathfrak{osp}(1,2n)$ is isomorphic to the
$\mathcal{W}B_{n}$-algebra introduced by Fateev and Lukyanov, and that the
$\mathcal{W}$-algebra associated with a subregular nilpotent element in
$\mathfrak{sl}_{n}$ is isomorphic to the $\mathcal{W}^{(2)}_{n}$-algebra
introduced by Feigin and Semikhatov.Comment: revised version, to appear in Sel. Math. New Se
Canonical bases of quantum Schubert cells and their symmetries

The goal of this work is to provide an elementary construction of the
canonical basis $\mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to
establish its invariance under modified Lusztig's symmetries. To that effect,
we obtain a direct characterization of the upper global basis $\mathbf B^{up}$
in terms of a suitable bilinear form and show that $\mathbf B(w)$ is contained
in $\mathbf B^{up}$ and its large part is preserved by modified Lusztig's
symmetries.Comment: AMSLaTeX, 32 pages,typos correcte
Almost regular Poisson manifolds and their holonomy groupoids

We look at Poisson geometry taking the viewpoint of singular foliations,
understood as suitable submodules generated by Hamiltonian vector fields rather
than partitions into (symplectic) leaves. The class of Poisson structures which
behave best from this point of view, are those whose submodule generated by
Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them
almost regular Poisson structures and determine them completely. They include
regular Poisson and log symplectic manifolds, as well as several other Poisson
structures whose symplectic foliation presents singularities.
  We show that the holonomy groupoid associated with an almost regular Poisson
structure is a Poisson groupoid, integrating a naturally associated Lie
bialgebroid. The Poisson structure on the holonomy groupoid is regular, and as
such it provides a desingularization. The holonomy groupoid is "minimal" among
Lie groupoids which give rise to the submodule generated by Hamiltonian vector
fields. This implies that, in the case of log-symplectic manifolds, the
holonomy groupoid coincides with the symplectic groupoid constructed by
Gualtieri and Li. Last, we focus on the integrability of almost regular Poisson
manifolds and exhibit the role of the second homotopy group of the
source-fibers of the holonomy groupoid.Comment: 36 pages. This version has been accepted for publicatio
Extremality and dynamically defined measures, part I: Diophantine
  properties of quasi-decaying measures

We present a new method of proving the Diophantine extremality of various
dynamically defined measures, vastly expanding the class of measures known to
be extremal. This generalizes and improves the celebrated theorem of Kleinbock
and Margulis ('98) resolving Sprind\v{z}uk's conjecture, as well as its
extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated
KLW. As applications we prove the extremality of all hyperbolic measures of
smooth dynamical systems with sufficiently large Hausdorff dimension, and of
the Patterson--Sullivan measures of all nonplanar geometrically finite groups.
The key technical idea, which has led to a plethora of new applications, is a
significant weakening of KLW's sufficient conditions for extremality.
  In Part I, we introduce and develop a systematic account of two classes of
measures, which we call $quasi$-$decaying$ and $weakly$ $quasi$-$decaying$. We
prove that weak quasi-decay implies strong extremality in the matrix
approximation framework (which has received much attention in recent years),
thus proving a conjecture of KLW. We also prove the "inherited exponent of
irrationality" version of this theorem, describing the relationship between the
Diophantine properties of certain subspaces of the space of matrices and
measures supported on these subspaces.
  In subsequent papers, we exhibit numerous examples of quasi-decaying
measures, in support of the thesis that "almost any measure from dynamics
and/or fractal geometry is quasi-decaying". In addition to the examples
described above, we also prove (for example) that Gibbs measures (including
conformal measures) of infinite iterated function systems are quasi-decaying,
even if the systems in question do not satisfy the open set condition. We also
discuss examples of non-extremal measures coming from dynamics, illustrating
where the theory must halt.Comment: Link to part II: arXiv:1508.0559
Affine Macdonald conjectures and special values of Felder-Varchenko
  functions

We refine the statement of the denominator and evaluation conjectures for
affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and prove the
first non-trivial cases of these conjectures. Our results provide a
q-deformation of the computation of genus 1 conformal blocks via elliptic
Selberg integrals by Felder-Stevens-Varchenko. They allow us to give precise
formulations for the affine Macdonald conjectures in the general case which are
consistent with computer computations.
  Our method applies recent work of the second named author to relate these
conjectures in the case of $U_q(\widehat{\mathfrak{sl}}_2)$ to evaluations of
certain theta hypergeometric integrals defined by Felder-Varchenko. We then
evaluate the resulting integrals, which may be of independent interest, by
well-chosen applications of the elliptic beta integral introduced by
Spiridonov.Comment: 26 pages. v3: minor edits for published versio