pinecone/core-2020-05-10-deduplication huggingface dataset:
Classification of finite-multiplicity symmetric pairs

We give a complete classification of the reductive symmetric pairs (G,H) for
which the homogeneous space $(G \times H)/diag(H)$ is real spherical in the
sense that a minimal parabolic subgroup has an open orbit.
  Combining with a criterion established in [T. Kobayashi--T. Oshima, Adv.
Math. 2013], we give a necessary and sufficient condition for a reductive
symmetric pair $(G,H)$ such that the multiplicities for the branching law of
the restriction any admissible smooth representation of $G$ to $H$ have
finiteness/boundedness property.Comment: To appear in Transformation Groups 19 (2014
Families of Group Actions, Generic Isotriviality, and Linearization

We study families of reductive group actions on A2 parametrized by curves and show that every faithful action of a non-finite reductive group on A3 is linearizable, i.e. G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine mor- phisms p: S → Y and q: T → Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant ́etale base change φ : U → Y . A special case is the following result. Call a morphism φ: X → Y a fibration with fiber F if φ is flat and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an ́etale dominant morphism μ: U → Y such that the pull-back is a trivial fiber bundle: U ×Y X ≃ U × F . As an application we give short proofs of the following two (known) results: (a) Every affine A1-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine A2-fibration over a smooth curve is locally trivial in the Zariski-topology
Deforming solutions of geometric variational problems with varying
  symmetry groups

We prove an equivariant implicit function theorem for variational problems
that are invariant under a varying symmetry group (corresponding to a bundle of
Lie groups). Motivated by applications to families of geometric variational
problems lacking regularity, several non-smooth extensions of the result are
discussed. Among such applications is the submanifold problem of deforming the
ambient metric preserving a given variational property of a prescribed family
of submanifolds, e.g., constant mean curvature, up to the action of the
corresponding ambient isometry groups.Comment: LaTeX2e, 26 pages, to appear in Transform. Group
Homogeneous spherical data of orbits in spherical embeddings

Let $G$ be a connected reductive complex algebraic group. Luna assigned to
any spherical homogeneous space $G/H$ a combinatorial object called a
homogeneous spherical datum. By a theorem of Losev, this object uniquely
determines $G/H$ up to $G$-equivariant isomorphism. In this paper, we determine
the homogeneous spherical datum of a $G$-orbit $X_0$ in a spherical embedding
$G/H \hookrightarrow X$. As an application, we obtain a description of the
colored fan associated to the spherical embedding $X_0 \hookrightarrow
\bar{X_0}$.Comment: 14 pages, 1 tabl
Extensions and block decompositions for finite-dimensional
  representations of equivariant map algebras

Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie
algebra $\mathfrak{g}$. The associated equivariant map algebra is the Lie
algebra of equivariant regular maps from $X$ to $\mathfrak{g}$. The irreducible
finite-dimensional representations of these algebras were classified in
previous work with P. Senesi, where it was shown that they are all tensor
products of evaluation representations and one-dimensional representations. In
the current paper, we describe the extensions between irreducible
finite-dimensional representations of an equivariant map algebra in the case
that $X$ is an affine scheme of finite type and $\mathfrak{g}$ is reductive.
This allows us to also describe explicitly the blocks of the category of
finite-dimensional representations in terms of spectral characters, whose
definition we extend to this general setting. Applying our results to the case
of generalized current algebras (the case where the group acting is trivial),
we recover known results but with very different proofs. For (twisted) loop
algebras, we recover known results on block decompositions (again with very
different proofs) and new explicit formulas for extensions. Finally,
specializing our results to the case of (twisted) multiloop algebras and
generalized Onsager algebras yields previously unknown results on both
extensions and block decompositions.Comment: 41 pages; v2: minor corrections, formatting changed to match
  published versio
Moduli spaces of (G,h)-constellations

Given an infinite reductive group G acting on an affine scheme X over C and a
Hilbert function h: Irr G \to N_0, we construct the moduli space M_{\theta}(X)
of \theta-stable (G,h)-constellations on X, which is a generalization of the
invariant Hilbert scheme after Alexeev and Brion and an analogue of the moduli
space of \theta-stable G-constellations for finite groups introduced by Craw
and Ishii. Our construction of a morphism M_{\theta}(X) \to X//G makes this
moduli space a candidate for a resolution of singularities of the quotient
X//G.Comment: 30 pages, published version and erratu
Harmonic structures and intrinsic torsion

We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form
is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds
that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our
constructions entail the notion of SO(4)-structures on 7-manifolds. We present
a thorough investigation of the intrinsic torsion of such structures, leading
to the construction of explicit Lie group examples with invariant intrinsic
torsion.Comment: 24 pages; v2, added two remarks concerning a potential analogous
  construction related to Spin(7) and the existence of a lattice in the
  solvable examples. To appear in Transformation Group
Pointed Hopf Actions On Fields, I

Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors in [18]. The answer turns out to be very simple–if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type.

The work begins by reducing to the case where H acts inner faithfully on a field; such a Hopf algebra is referred to as Galois-theoretical. We present examples of such Hopf algebras, which include the Taft algebras, uq(sl₂), and some Drinfeld twists of other small quantum groups. We also give many examples of finite dimensional Hopf algebras which are not Galois-theoretical. Classification results on finite dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type will be provided in the sequel, Part II, of this study.National Science Foundation (U.S.) (DMS-1000173)National Science Foundation (U.S.) (DMS-1102548)National Science Foundation (U.S.) (DMS-1401207
Quantum loop algebras and l-root operators

This is the accepted manuscript of the following article: Charles Young, ???Quantum loop algebras and l-root operators???, Transformation Groups, Vol. 20(4): 1195-1226, September 2015. The final published version is available at: https://link.springer.com/article/10.1007%2Fs00031-015-9339-4 ?? Springer Science+Business Media New York (2015)Let g be a simple Lie algebra and q transcendental. We consider the category C_P of finite-dimensional representations of the quantum loop algebra Uq(Lg) in which the poles of all l-weights belong to specified finite sets P. Given the data (g,q,P), we define an algebra A whose raising/lowering operators are constructed to act with definite l-weight (unlike those of Uq(Lg) itself). It is shown that there is a homomorphism Uq(Lg) -> A such that every representation V in C_P is the pull-back of a representation of A
Invariant Functionals on the Speh representation

We study Sp(2n,R)-invariant functionals on the spaces of smooth vectors in
Speh representations of GL(2n,R).
  For even n we give expressions for such invariant functionals using an
explicit realization of the space of smooth vectors in the Speh
representations. Furthermore, we show that the functional we construct is, up
to a constant, the unique functional on the Speh representation which is
invariant under the Siegel parabolic subgroup of Sp(2n,R). For odd n we show
that the Speh representations do not admit an invariant functional with respect
to the subgroup U(n) of Sp(2n,R) consisting of unitary matrices.
  Our construction, combined with the argument in [GOSS12], gives a purely
local and explicit construction of Klyachko models for all unitary
representations of GL(2n,R).Comment: 14 pages. v4: minor corrections in Theorem 2.2, Lemma 2.9 and section
Lifting differentiable curves from orbit spaces

Let $\rho : G \rightarrow \operatorname{O}(V)$ be a real finite dimensional
orthogonal representation of a compact Lie group, let $\sigma =
(\sigma_1,\ldots,\sigma_n) : V \to \mathbb R^n$, where
$\sigma_1,\ldots,\sigma_n$ form a minimal system of homogeneous generators of
the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg}
\sigma_i$. We prove that for each $C^{d-1,1}$-curve $c$ in $\sigma(V) \subseteq
\mathbb R^n$ there exits a locally Lipschitz lift over $\sigma$, i.e., a
locally Lipschitz curve $\overline c$ in $V$ so that $c = \sigma \circ
\overline c$, and we obtain explicit bounds for the Lipschitz constant of
$\overline c$ in terms of $c$. Moreover, we show that each $C^d$-curve in
$\sigma(V)$ admits a $C^1$-lift. For finite groups $G$ we deduce a
multivariable version and some further results.Comment: 25 pages; section on orbit spaces as differentiable spaces added,
  some typos corrected; accepted for publication in Transformation Group
Jordan-Kronecker invariants of finite-dimensional Lie algebras

The final publication is available at Springer via http://dx.doi.org/10.1007/s00031-015-9353-6For any finite-dimensional Lie algebra we introduce the notion of Jordan–Kronecker invariants, study their properties, and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko–Fomenko’s argument shift method. We also state a generalised argument shift conjecture and prove it for many series of Lie algebras
Sheets of symmetric Lie algebras and slice induction

23 pagesInternational audienceIn this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more details the sheets of the non-trivial symmetric Lie algebra of type G2. We characterize their singular loci and provide a nice desingularization lying in so7.In this new version, computations of section 4 are pared down. Important modifications of the exposition of Section 3 on slice induction
Hamiltonian circle actions on eight dimensional manifolds with minimal
  fixed sets

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic
manifold $M$ with exactly five fixed points, which is the smallest possible
fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$
satisfies an extra "positivity condition" then the isotropy weights at the
fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$.
Therefore, the (equivariant) cohomology rings and the (equivariant) Chern
classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z})
\simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove that
this positivity condition always holds for these manifolds. This completes the
proof of the "symplectic Petrie conjecture" for Hamiltonian circle actions on
on 8-dimensional closed symplectic manifolds with minimal fixed sets.Comment: To appear in Transformation Group
Newton-Okounkov polytopes of flag varieties

We compute the Newton--Okounkov bodies of line bundles on the complete flag
variety of GL_n for a geometric valuation coming from a flag of translated
Schubert subvarieties. The Schubert subvarieties correspond to the terminal
subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of
the longest element in the Weyl group. The resulting Newton--Okounkov bodies
coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.Comment: 16 pages, 2 figures, final version, typos corrected, details added, a
  new proof (Example 2.9, Remark 2.10) outline
Bott-Borel-Weil theory and Bernstein-Gel'Fand-Gel'Fand reciprocity for the Lie superalgebras

The main focus of this paper is Bott-Borel-Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for (m|2), D(2, 1; alpha), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite-dimensional (m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules
Polynomials for symmetric orbit closures in the flag variety

In [Wyser-Yong '13] we introduced polynomial representatives of cohomology
classes of orbit closures in the flag variety, for the symmetric pair
$(GL_{p+q}, GL_p \times GL_q)$. We present analogous results for the remaining
symmetric pairs of the form $(GL_n,K)$, i.e., $(GL_n,O_n)$ and
$(GL_{2n},Sp_{2n})$. We establish the well-definedness of certain
representatives from [Wyser '13]. It is also shown that the representatives
have the combinatorial properties of nonnegativity and stability. Moreover, we
give some extensions to equivariant $K$-theory.Comment: 22 pages, 3 figures, 5 tables. Results from V1 have been extended
  significantl
Equivariant compactifications of a nilpotent group by $G/P$

Let $G$ be a simple complex algebraic group, $P$ a parabolic subgroup of $G$
and $N$ the unipotent radical of $P.$ The so-called equivariant
compactification of $N$ by $G/P$ is given by an action of $N$ on $G/P$ with a
dense open orbit isomorphic to $N$. In this article, we investigate how many
such equivariant compactifications there exist. Our result says that there is a
unique equivariant compactification of $N$ by $G/P$, up to isomorphism, except
$\P^n$.Comment: 20 pages, to appear in Transformation Group
Root systems and symmetries of torus manifolds

We associate a root system to a finite set in a free abelian group and prove
that its irreducible subsystem is of type A, B or D. We apply this general
result to a torus manifold, where a torus manifold is a $2n$-dimensional
connected closed smooth manifold with a smooth effective action of an
$n$-dimensional compact torus having a fixed point, and show that if the torus
action extends to a smooth action of a connected compact Lie group $G$, then a
simple factor of the Lie algebra of $G$ is of type A, B or D. This gives an
alternative proof to Wiemeler's theorem. We also discuss a similar problem for
a torus manifold with an invariant stably complex structure. In this case only
type A appears.Comment: 21 pages, v2: deleted Lemma 3.11, added Remark 4.7, references
  update
On the orbits of a Borel subgroup in abelian ideals

Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let
$\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal $\mathfrak
a$ is determined by certain subset $\Delta_{\mathfrak a}$ of positive roots,
and using $\Delta_{\mathfrak a}$ we give an explicit classification of the
$B$-orbits in $\mathfrak a$ and $\mathfrak a^*$. Our description visibly
demonstrates that there are finitely many $B$-orbits in both cases. We also
describe the Pyasetskii correspondence between the $B$-orbits in $\mathfrak a$
and $\mathfrak a^*$ and the invariant algebras $\Bbbk[\mathfrak a]^U$ and
$\Bbbk[\mathfrak a^*]^U$, where $U=(B,B)$. As an application, the number of
$B$-orbits in the abelian nilradicals is computed. We also discuss related
results of A.Melnikov and others for classical groups and state a general
conjecture on the closure and dimension of the $B$-orbits in the abelian
nilradicals, which exploits a relationship between between $B$-orbits and
involutions in the Weyl group.Comment: 24 page
Twisted forms of toric varieties

We consider the set of forms of a toric variety over an arbitrary field:
those varieties which become isomorphic to a toric variety after base field
extension. In contrast to most previous work, we also consider arbitrary
isomorphisms rather than just those that respect a torus action. We define an
injective map from the set of forms of a toric variety to a non-abelian second
cohomology set, which generalizes the usual Brauer class of a Severi-Brauer
variety. Additionally, we define a map from the set of forms of a toric variety
to the set of forms of a separable algebra along similar lines to a
construction of A. Merkurjev and I. Panin. This generalizes both a result of
M.~Blunk for del Pezzo surfaces of degree 6, and the standard bijection between
Severi-Brauer varieties and central simple algebrasComment: 41 pages; numerous revisions: introduction more accessible, results
  now weaker for singular varieties in positive characteristi
Equivariant Chow classes of matrix orbit closures

Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the
$G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$
matrices is determined by a matroid. To do this, we split the natural
surjective map from the $G$ equvariant Chow ring of the space of matrices to
the torus equivariant Chow ring of the Grassmannian. The splitting takes the
class of a Schubert variety to the corresponding factorial Schur polynomial,
and also has the property that the class of a subvariety of the Grassmannian is
mapped to the class of the closure of those matrices whose row span is in the
variety.Comment: 11 pages. Theorem 3.5 here proves a version of the main result of
  arXiv:1306.1810v5, the proof of which contained an erro
Comparison of canonical bases for Schur and universal enveloping
  algebras

We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur
algebra are compatible; in fact we extend this result to $p$-canonical bases.
This follows immediately from a fullness result from a functor categorifying
this map. In order to prove this result, we also explain the connections
between categorifications of the Schur algebra which arise from parity sheaves
on partial flag varieties, singular Soergel bimodules and Khovanov and Lauda's
"flag category," which are of some independent interest.Comment: 12 pages. v2: correcting typo
Geometric structures modeled on smooth projective horospherical
  varieties of Picard number one

Geometric structures modeled on rational homogeneous manifolds are studied to
characterize rational homogeneous manifolds and to prove their deformation
rigidity. To generalize these characterizations and deformation rigidity
results to quasihomogeneous varieties, we first study horospherical varieties
and geometric structures modeled on horospherical varieties. Using Cartan
geometry, we prove that a geometric structure modeled on a smooth projective
horospherical variety of Picard number one is locally equivalent to the
standard geometric structure when the geometric structure is defined on a Fano
manifold of Picard number one.Comment: 32 page
A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric
  Macdonald polynomials at $t=0$ and Demazure characters

We establish the equality of the specialization $E_{w\lambda}(x;q,0)$ of the
nonsymmetric Macdonald polynomial $E_{w\lambda}(x;q,t)$ at $t=0$ with the
graded character $\mathop{\rm gch} U_{w}^{+}(\lambda)$ of a certain
Demazure-type submodule $U_{w}^{+}(\lambda)$ of a tensor product of
"single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie
algebra, where $\lambda$ is a dominant integral weight and $w$ is a (finite)
Weyl group element, this generalizes our previous result, that is, the equality
between the specialization $P_{\lambda}(x;q,0)$ of the symmetric Macdonald
polynomial $P_{\lambda}(x;q,t)$ at $t=0$ and the graded character of a tensor
product of single-column Kirillov--Reshetikhin modules. We also give two
combinatorial formulas for the mentioned specialization of a nonsymmetric
Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the
other in terms of the quantum alcove model.Comment: updated reference
On the irreducible components of moduli schemes for affine spherical
  varieties

We give a combinatorial description of all affine spherical varieties with
prescribed weight monoid $\Gamma$. As an application, we obtain a
characterization of the irreducible components of Alexeev and Brion's moduli
scheme $\mathrm M_\Gamma$ for such varieties. Moreover, we find several
sufficient conditions for $\mathrm M_\Gamma$ to be irreducible and exhibit
several examples where $\mathrm M_\Gamma$ is reducible. Finally, we provide
examples of non-reduced $\mathrm M_\Gamma$.Comment: v4: 26 pages, final versio
On bases of some simple modules of symmetric groups and Hecke algebras

We consider simple modules for a Hecke algebra with a parameter of quantum
characteristic $e$. Equivalently, we consider simple modules $D^{\lambda}$,
labelled by $e$-restricted partitions $\lambda$ of $n$, for a cyclotomic KLR
algebra $R_n^{\Lambda_0}$ over a field of characteristic $p\ge 0$, with mild
restrictions on $p$. If all parts of $\lambda$ are at most $2$, we identify a
set $\mathsf{DStd}_{e,p}(\lambda)$ of standard $\lambda$-tableaux, which is
defined combinatorially and naturally labels a basis of $D^{\lambda}$. In
particular, we prove that the $q$-character of $D^{\lambda}$ can be described
in terms of $\mathsf{DStd}_{e,p}(\lambda)$. We show that a certain natural
approach to constructing a basis of an arbitrary $D^{\lambda}$ does not work in
general, giving a counterexample to a conjecture of Mathas.Comment: Final version, to appear in Transform. Group
Virtualization map for the Littelmann path model

We show the natural embedding of weight lattices from a diagram folding is a
virtualization map for the Littelmann path model, which recovers a result of
Kashiwara. As an application, we give a type independent proof that certain
Kirillov--Reshetikhin crystals respect diagram foldings, which is a known
result on a special case of a conjecture given by Okado, Schilling, and
Shimozono.Comment: 14 pages, 1 figure; corrected statement of Theorem 4.2 in v3; minor
  corrections from referee report in v