1,176 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Diophantine problem in Chevalley groups
In this paper we study the Diophantine problem in Chevalley groups , where is an indecomposable root system of rank , is
an arbitrary commutative ring with . We establish a variant of double
centralizer theorem for elementary unipotents . This theorem is
valid for arbitrary commutative rings with . The result is principle to show
that any one-parametric subgroup , , is Diophantine
in . Then we prove that the Diophantine problem in is
polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine
problem in . This fact gives rise to a number of model-theoretic corollaries
for specific types of rings.Comment: 44 page
Deformation theory of G-valued pseudocharacters and symplectic determinant laws
We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and
Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are
very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group,
where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are
so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a
generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion
of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters.
We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant
laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from
small characteristics.
We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously,
we obtain that tensor products of determinant laws exist and are compatible with tensor products
of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor
product of Taylor’s pseudocharacters.
We generalize some of the results of [Che14] to general reductive groups, in particular we show that
the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely
generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show,
that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Φ_p.
One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp
be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and
concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater
of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension
n(2n + 1)[F : Qp].
We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected
reductive groups G come from a semisimple representation. We introduce a formal scheme
and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on
our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition
as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to
conjugacy and Frobenius automorphisms.
At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic
groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under
an involution. We prove the expected basic properties in that we show that symplectic determinant laws
over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and
that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings
are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s
pseudocharacters and show that it is an isomorphism over reduced rings
Cohomology rings of extended powers and free infinite loop spaces
We calculate mod-p cohomology of extended powers, and their group completions
which are free infinite loop spaces. We consider the cohomology of all extended
powers of a space together and identify a Hopf ring structure with divided
powers within which cup product structure is more readily computable than on
its own. We build on our previous calculations of cohomology of symmetric
groups, which are the cohomology of extended powers of a point, the well-known
calculation of homology, and new results on cohomology of symmetric groups with
coefficients in the sign representation. We then use this framework to
understand cohomology rings of related spaces such as infinite extended powers
and free infinite loop spaces.Comment: 37 pages, 1 figur
Automorphisms of del Pezzo surfaces in odd characteristic
We complete the classification of automorphism groups of del Pezzo surfaces
over algebraically closed fields of odd positive characteristic.Comment: 35 pages, comments welcom
Induced log-concavity of equivariant matroid invariants
Inspired by the notion of equivariant log-concavity, we introduce the concept
of induced log-concavity for a sequence of representations of a finite group.
For an equivariant matroid equipped with a symmetric group action or a finite
general linear group action, we transform the problem of proving the induced
log-concavity of matroid invariants to that of proving the Schur positivity of
symmetric functions. We prove the induced log-concavity of the equivariant
Kazhdan-Lusztig polynomials of -niform matroids equipped with the action of
a finite general linear group, as well as that of the equivariant
Kazhdan-Lusztig polynomials of uniform matroids equipped with the action of a
symmetric group.
As a consequence of the former, we obtain the log-concavity of
Kazhdan-Lusztig polynomials of -niform matroids, thus providing further
positive evidence for Elias, Proudfoot and Wakefield's log-concavity conjecture
on the matroid Kazhdan-Lusztig polynomials. From the latter we obtain the
log-concavity of Kazhdan-Lusztig polynomials of uniform matroids, which was
recently proved by Xie and Zhang by using a computer algebra approach. We also
establish the induced log-concavity of the equivariant characteristic
polynomials and the equivariant inverse Kazhdan-Lusztig polynomials for
-niform matroids and uniform matroids.Comment: 36 page
Lie algebra actions on module categories for truncated shifted Yangians
We develop a theory of parabolic induction and restriction functors relating
modules over Coulomb branch algebras, in the sense of
Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's
induction and restriction functors for Cherednik algebras, but their definition
uses different tools.
After this general definition, we focus on quiver gauge theories attached to
a quiver . The induction and restriction functors allow us to define a
categorical action of the corresponding symmetric Kac-Moody algebra
on category for these Coulomb branch
algebras. When is of Dynkin type, the Coulomb branch algebras are
truncated shifted Yangians and quantize generalized affine Grassmannian slices.
Thus, we regard our action as a categorification of the geometric Satake
correspondence.
To establish this categorical action, we define a new class of "flavoured"
KLRW algebras, which are similar to the diagrammatic algebras originally
constructed by the second author for the purpose of tensor product
categorification. We prove an equivalence between the category of
Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a
flavoured KLRW algebra. This equivalence relates the categorical action by
induction and restriction functors to the usual categorical action on modules
over a KLRW algebra.Comment: 66 pages, version 2: many corrections, improved treatment of GK
dimension, 71 page
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