pinecone/core-2020-05-10-deduplication huggingface dataset:
Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be one of the complex Lie groups $O_M$ and $Sp_M$. The irreducible
finite-dimensional representations of the group $G_M$ are labeled by partitions
$\mu$ satisfying certain extra conditions. Let $U$ be the representation of
$G_M$ corresponding to $\mu$. Regard the direct product $G_N\times G_M$ as a
subgroup of $G_{N+M}$. Let $V$ be the irreducible representation of $G_{N+M}$
corresponding to a partition $\lambda$. Consider the vector space
$W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $G_N$. Let $n$
be sum of parts of $\lambda$ less the sum of parts of $\mu$. For any choice of
a standard Young tableau of skew shape $\lambda/\mu$, we realize $W$ as a
subspace in the tensor product of $n$ copies of the defining $N$-dimensional
representation of $G_N$. This subspace is determined as the image of a certain
linear operator $F(M)$ in the tensor product, given by an explicit formula.
When M=0 and $W=V$ is an irreducible representation of $G_N$, we recover the
classical realization of $V$ as a subspace in the space of all traceless
tensors. Then the operator F(0) can be regarded as the analogue for $G_N$ of
the Young symmetrizer, corresponding to the chosen standard tableau of shape
$\lambda$. Even in the special case M=0, our formula for the operator $F(M)$ is
new. Our results are applications of representation theory of the twisted
Yangian, corresponding to $G_N$. In particular, $F(M)$ is an intertwining
operator between two representations of the twisted Yangian in the $n$-fold
tensor product.Comment: 60 pages; final version, Section 0 adde
Linearizability of d-webs, d ≥ 4, on
two-dimensional manifolds

Dette er forfatternes aksepterte versjon.We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a
two-dimensional manifold and prove that their vanishing is necessary and
sufficient for a d-web to be linearizable. If one writes the above invariants
in terms of web functions f(x, y) and g4(x, y), ..., gd(x, y), then necessary
and sufficient conditions for the linearizabilty of a d-web are two PDEs of
the fourth order with respect to f and g4, and d − 4 PDEs of the second
order with respect to f and g4, ..., gd. For d = 4, this result confirms
Blaschke’s conjecture on the nature of conditions for the linearizabilty
of a 4-web. We also give Mathematica codes for testing 4- and d-webs
(d > 4) for linearizability and examples of their usage
Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational
maps to a given variety $X$, with value in an associated Chow group.
Applications include new birational invariants; comparison results for Chern
classes and numbers of nonsingular birational varieties; `stringy' Chern
classes of singular varieties; and a zeta function specializing to the
topological zeta function.
  In its simplest manifestation, the integral gives a new expression for
Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them
into a context in which a `change-of-variable' formula holds. v2: References
added, and overly optimistic claims concerning non log-terminal singularities
expunged.Comment: 42 pages, LaTeX, 2 figure
On the degree of Polar Transformations -- An approach through
  Logarithmic Foliations

We investigate the degree of the polar transformations associated to a
certain class of multi-valued homogeneous functions. In particular we prove
that the degree of the pre-image of generic linear spaces by a polar
transformation associated to a homogeneous polynomial $F$ is determined by the
zero locus of $F$. For zero dimensional-dimensional linear spaces this was
conjecture by Dolgachev and proved by Dimca-Papadima using topological
arguments. Our methods are algebro-geometric and rely on the study of the Gauss
map of naturally associated logarithmic foliations
Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation $u_t=Lu$ where $L$ is a second-order difference
operator in a discrete variable $n$. The fundamental solution has an expansion
in terms of the Bessel functions of imaginary argument. The coefficients
$\alpha_k(n,m)$ in this expansion are analogs of Hadamard's coefficients for
the (continuous) Schrodinger operator.
  We derive an explicit formula for $\alpha_k$ in terms of the wave and the
adjoint wave functions of the Toda lattice hierarchy. As a first application of
this result, we prove that the values of these coefficients on the diagonals
$n=m$ and $n=m+1$ define a hierarchy of differential-difference equations which
is equivalent to the Toda lattice hierarchy. Using this fact and the
correspondence between commutative rings of difference operators and algebraic
curves we show that the fundamental solution can be summed up, giving a finite
formula involving only two Bessel functions with polynomial coefficients in the
time variable $t$, if and only if the operator $L$ belongs to the family of
bispectral operators constructed in [18].Comment: corrected typo
Twisted Whittaker Model and Factorizable Sheaves

Let \(G\) be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group \(\check G\) in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian \(Gr_G=G((t))/G[[t]]\) of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to \(\check G\) in terms of the geometry of \(Gr_G\). The idea of the construction belongs to Jacob Lurie.Mathematic
Non-linear Lie conformal algebras with three generators

We classify certain non-linear Lie conformal algebras with three generators,
which can be viewed as deformations of the current Lie conformal algebra of
sl_2. In doing so we discover an interesting 1-parameter family of non-linear
Lie conformal algebras R_{-1}^d and the corresponding freely generated vertex
algebras V_{-1}^d, which includes for d=1 the affine vertex algebra of sl_2 at
the critical level k=-2. We construct free-field realizations of the algebras
V_{-1}^d extending the Wakimoto realization of the affine vertex algebra of
sl_2 at the critical level, and we compute their Zhu algebras.Comment: 36 pages, v2 minor change
Parking functions and vertex operators

We introduce several associative algebras and series of vector spaces
associated to these algebras. Using lattice vertex operators, we obtain
dimension and character formulae for these spaces. In particular, we a series
of representations of symmetric groups which turn out to be isomorphic to
parking function modules. We also construct series of vector spaces whose
dimensions are Catalan numbers and Fuss--Catalan numbers respectively.
Conjecturally, these spaces are related to spaces of global sections of vector
bundles on (zero fibres of) Hilbert schemes and representations of rational
Cherednik algebras.Comment: 17 pages, 1 figure, proofs are made more detailed. To appear in
  Selecta Mathematic
Vertex operators and the geometry of moduli spaces of framed
  torsion-free sheaves

We define complexes of vector bundles on products of moduli spaces of framed
rank r torsion-free sheaves on the complex projective plane. The top
non-vanishing Chern classes of the cohomology of these complexes yield actions
of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology
of the moduli spaces. In this way we obtain a geometric realization of the
boson-fermion correspondence and related vertex operators.Comment: 36 pages; v2: Definition of geometric Heisenberg operators modified;
  v3: Minor typos correcte
Parabolic induction and restriction functors for rational Cherednik algebras

Parabolic induction and restriction functors play an important role in the
representation theory of finite and affine Hecke algebras. This makes it desirable
to generalize them to the setting of double affine Hecke algebras, or
Cherednik algebras. However, a naive attempt to do so fails: the definition
of parabolic induction and restriction functors for finite and affine Hecke
algebras uses the fact that the Hecke algebra attached to a parabolic subgroup
can be embedded into the Hecke algebra attached to the whole group,
which is not the case in the double affine setting
Open orbifold Gromov-Witten invariants of [C^3/Z_n]: localization and
  mirror symmetry

We develop a mathematical framework for the computation of open orbifold
Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with
predictions from open string mirror symmetry. To this aim we set up a
computation of open string invariants in the spirit of Katz-Liu, defining them
by localization. The orbifold is viewed as an open chart of a global quotient
of the resolved conifold, and the Lagrangian as the fixed locus of an
appropriate anti-holomorphic involution. We consider two main applications of
the formalism. After warming up with the simpler example of [C^3/Z_3], where we
verify physical predictions of Bouchard, Klemm, Marino and Pasquetti, the main
object of our study is the richer case of [C^3/Z_4], where two different
choices are allowed for the Lagrangian. For one choice, we make numerical
checks to confirm the B-model predictions; for the other, we prove a mirror
theorem for orbifold disc invariants, match a large number of annulus
invariants, and give mirror symmetry predictions for open string invariants of
genus \leq 2.Comment: 44 pages + appendices; v2: exposition improved, misprints corrected,
  version to appear on Selecta Mathematica; v3: last minute mistake found and
  fixed for the symmetric brane setup of [C^3/Z_4]; in pres
On the lifting of the Nagata automorphism

It is proved that all wild z-automorphisms including the well-known Nagata automorphism (all wild z-coordinates including the Nagata coordinates, respectively) of the polynomial algebra F[x, y, z] over an arbitrary field F cannot be lifted to a z-automorphism (z-coordinate, respectively) of the free associative algebra 〈x, y, z〉. The proof is based on the following two new results, which have their own interests: degree estimate of Q *F F〈x 1, ..., x n〉 and tameness of the automorphism group Aut q(Q * F F〈x, y〉). The structure of the group of all z-automorphisms of the free associative algebra F〈x, y〉 over an arbitrary field F is also determined. © 2011 Springer Basel AG.postprin
Chiral Koszul Duality

We extend the theory of chiral and factorization algebras, developed for
curves by Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004), to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen’s homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004) on chiral enveloping algebras of \(\star\)-Lie algebras.Mathematic
Cluster structures on quantum coordinate rings

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a
symmetric Kac-Moody group G associated with a Weyl group element w has the
structure of a quantum cluster algebra. This quantum cluster structure arises
naturally from a subcategory C_w of the module category of the corresponding
preprojective algebra. An important ingredient of the proof is a system of
quantum determinantal identities which can be viewed as a q-analogue of a
T-system. In case G is a simple algebraic group of type A, D, E, we deduce from
these results that the quantum coordinate ring of an open cell of a partial
flag variety attached to G also has a cluster structure.Comment: v2: minor corrections. v3: references updated, final version to
  appear in Selecta Mathematic
Homology of Littlewood complexes

Let V be a symplectic vector space of dimension 2n. Given a partition \lambda
with at most n parts, there is an associated irreducible representation
S_{[\lambda]}(V) of Sp(V). This representation admits a resolution by a natural
complex L^\lambda, which we call the Littlewood complex, whose terms are
restrictions of representations of GL(V). When \lambda has more than n parts,
the representation S_{[\lambda]}(V) is not defined, but the Littlewood complex
L^\lambda still makes sense. The purpose of this paper is to compute its
homology. We find that either L^\lambda is acyclic or that it has a unique
non-zero homology group, which forms an irreducible representation of Sp(V).
The non-zero homology group, if it exists, can be computed by a rule
reminiscent of that occurring in the Borel-Weil-Bott theorem. This result can
be interpreted as the computation of the "derived specialization" of
irreducible representations of Sp(\infty), and as such categorifies earlier
results of Koike-Terada on universal character rings. We prove analogous
results for orthogonal and general linear groups. Along the way, we will see
two topics from commutative algebra: the minimal free resolutions of
determinantal ideals and Koszul homology.Comment: 32 pages; v2: corrected minor typos and changed theorem/equation
  numberin
Discrete subgroups of locally definable groups

We work in the category of locally definable groups in an o-minimal expansion
of a field. Eleftheriou and Peterzil conjectured that every definably generated
abelian connected group G in this category is a cover of a definable group. We
prove that this is the case under a natural convexity assumption inspired by
the same authors, which in fact gives a necessary and sufficient condition. The
proof is based on the study of the zero-dimensional compatible subgroups of G.
Given a locally definable connected group G (not necessarily definably
generated), we prove that the n-torsion subgroup of G is finite and that every
zero-dimensional compatible subgroup of G has finite rank. Under a convexity
hypothesis we show that every zero-dimensional compatible subgroup of G is
finitely generated.Comment: Final version. 17 pages. To appear in Selecta Mathematic
Constructible nabla-modules on curves

39 pagesInternational audienceLet $\mathcal V$ be a discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal V$. We introduce the notion of constructible convergent $\nabla$-module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fibre of $X$. A constructible module is an $\mathcal O_{X_{K}^{\mathrm{an}}}$-module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$. The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla$-modules to the category of $\mathcal D^\dagger_{\hat X \mathbf Q}$-modules. We show that if $X$ is endowed with a lifting of the absolute Frobenius of $X$, then specialization induces an equivalence between constructible $F$-$\nabla$-modules and perverse holonomic $F$-$\mathcal D^\dagger_{\hat X \mathbf Q}$-modules
Noncommutative Artin motives

In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)[subscript Q] AM ( k ) Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives, we then show that the image of this full embedding, which we call the category NAM(k)[subscript Q] NAM ( k ) Q of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group Gal(k−/k) Gal ( k ¯ / k ) from the Tannakian formalism applied to NAM(k)[subscript Q] NAM ( k ) Q . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change, we then construct a short exact sequence relating Gal(k−/k) Gal ( k ¯ / k ) with the noncommutative motivic Galois groups of k and k− k ¯ . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne–Milne. In the mixed world, we introduce the triangulated category NMAM(k)[subscript Q] NMAM ( k ) Q of noncommutative mixed Artin motives and construct a faithful functor from the classical category MAM(k)[subscript Q] MAM ( k ) Q of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero NMAM(k)[subscript Q] NMAM ( k ) Q is much richer than MAM(k)[subscript Q] MAM ( k ) Q since its higher Ext-groups encode all the (rationalized) higher algebraic K K -theory of finite étale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne–Milne’s short exact sequence.17 Jul 2012 Original ManuscriptNEC Corporation Fund for Research in Computers and Communications (Award-2742738
Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper we generalize the construction of the nil Hecke ring of
Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology
theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's
K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The
resulting object, which we call a formal (affine) Demazure algebra, is
parameterized by a one-dimensional commutative formal group law and has the
following important property: specialization to the additive and multiplicative
periodic formal group laws yields completions of the nil Hecke and the 0-Hecke
rings respectively. We also introduce a deformed version of the formal (affine)
Demazure algebra, which we call a formal (affine) Hecke algebra. We show that
the specialization of the formal (affine) Hecke algebra to the additive and
multiplicative periodic formal group laws gives completions of the degenerate
(affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We
show that all formal affine Demazure algebras (and all formal affine Hecke
algebras) become isomorphic over certain coefficient rings, proving an analogue
of a result of Lusztig.Comment: 28 pages. v2: Some results strengthened and references added. v3:
  Minor corrections, section numbering changed to match published version. v4:
  Sign errors in Proposition 6.8(d) corrected. This version incorporates an
  erratum to the published versio
Quantum cluster characters of Hall algebras

The aim of the present paper is to introduce a generalized quantum cluster
character, which assigns to each object V of a finitary Abelian category C over
a finite field FF_q and any sequence ii of simple objects in C the element
X_{V,ii} of the corresponding algebra P_{C,ii} of q-polynomials. We prove that
if C was hereditary, then the assignments V-> X_{V,ii} define algebra
homomorphisms from the (dual) Hall-Ringel algebra of C to the P_{C,ii}, which
generalize the well-known Feigin homomorphisms from the upper half of a quantum
group to q-polynomial algebras.
  If C is the representation category of an acyclic valued quiver (Q,d) and
ii=(ii_0,ii_0), where ii_0 is a repetition-free source-adapted sequence, then
we prove that the ii-character X_{V,ii} equals the quantum cluster character
X_V introduced earlier by the second author in [29] and [30]. Using this
identification, we deduce a quantum cluster structure on the quantum unipotent
cell corresponding to the square of a Coxeter element. As a corollary, we prove
a conjecture from the joint paper [5] of the first author with A. Zelevinsky
for such quantum unipotent cells. As a byproduct, we construct the quantum
twist and prove that it preserves the triangular basis introduced by A.
Zelevinsky and the first author in [6].Comment: AMS LaTeX, 46 pages, a reference added, to appear in Selecta
  Mathematic
Quivers with potentials associated to triangulated surfaces, part IV:
  Removing boundary assumptions

We prove that the quivers with potentials associated to triangulations of
surfaces with marked points, and possibly empty boundary, are non-degenerate,
provided the underlying surface with marked points is not a closed sphere with
exactly 5 punctures. This is done by explicitly defining the QPs that
correspond to tagged triangulations and proving that whenever two tagged
triangulations are related by a flip, their associated QPs are related by the
corresponding QP-mutation. As a byproduct, for (arbitrarily punctured) surfaces
with non-empty boundary we obtain a proof of the non-degeneracy of the
associated QPs which is independent from the one given by the author in the
first paper of the series.
  The main tool used to prove the aforementioned compatibility between flips
and QP-mutations is what we have called \emph{Popping Theorem}, which, roughly
speaking, says that an apparent lack of symmetry in the potentials arising from
ideal triangulations with self-folded triangles can be fixed by a suitable
right-equivalence.Comment: v5: Final version. Published by Selecta Mathematica (New Series).
  Published version contains a few minor redaction errors. E.g., "related by a
  flip" and "related by a QP-mutation" were incorrectly replaced by "related to
  a flip" and "related to a QP-mutation" therein. Changes following suggestions
  by the referee. 36 pages, 21 figures. Dedicated to the memory of Professor
  Andrei Zelevinsk
Noetherianity of some degree two twisted commutative algebras

In recent years, researchers have discovered various large algebraic
structures that have surprising finiteness properties, such as FI-modules and
Delta-modules. In this paper, we add another example to the growing list: we
show that certain degree two twisted commutative algebras are noetherian. This
example appears to have some fundamental differences from previous examples,
and is therefore especially interesting. Reflective of this, our proof
introduces new methods for establishing noetherianity that are likely to be
applicable in other situations. The algebras considered in this paper are
closely related to the stable representation theory of classical groups, which
is one source of motivation for their study.Comment: 21 pages; v2: small corrections and added Example 1.
Instanton Floer homology and contact structures

We define an invariant of contact 3-manifolds with convex boundary using
Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of
our knowledge, this is the first invariant of contact manifolds -- with or
without boundary -- defined in the instanton Floer setting. We prove that our
invariant vanishes for overtwisted contact structures and is nonzero for
contact manifolds with boundary which embed into Stein fillable contact
manifolds. Moreover, we propose a strategy by which our contact invariant might
be used to relate the fundamental group of a closed contact 3-manifold to
properties of its Stein fillings. Our construction is inspired by a
reformulation of a similar invariant in the monopole Floer setting defined by
the authors in [1].Comment: 32 pages, 7 figures; this paper was originally part of
  arXiv:1403.193