1,249 research outputs found

### Gelfand-Tsetlin modules in the Coulomb context

This paper gives a new perspective on the theory of principal Galois orders
as developed by Futorny, Ovsienko, Hartwig and others. Every principal Galois
order can be written as $eFe$ for any idempotent $e$ in an algebra $F$, which
we call a flag Galois order; and in most important cases we can assume that
these algebras are Morita equivalent. These algebras have the property that the
completed algebra controlling the fiber over a maximal ideal has the same form
as a subalgebra in a skew group ring, which gives a new perspective to a number
of result about these algebras.
We also discuss how this approach relates to the study of Coulomb branches in
the sense of Braverman-Finkelberg-Nakajima, which are particularly beautiful
examples of principal Galois orders. These include most of the interesting
examples of principal Galois orders, such as $U(\mathfrak{gl}_n)$. In this
case, all the objects discussed have a geometric interpretation which endows
the category of Gelfand-Tsetlin modules with a graded lift and allows us to
interpret the classes of simple Gelfand-Tsetlin modules in terms of dual
canonical bases for the Grothendieck group. In particular, we classify
Gelfand-Tsetlin modules over $U(\mathfrak{gl}_n)$ and relate their characters
to a generalization of Leclerc's shuffle expansion for dual canonical basis
vectors.
Finally, as an application, we confirm a conjecture of Mazorchuk, showing
that the weights of the Gelfand-Tsetlin integrable system which appear in
finite-dimensional modules never appear in an infinite-dimensional simple
module.Comment: 37 pages; v3: minor improvements before submissio

### Geometry and categorification

We describe a number of geometric contexts where categorification appears
naturally: coherent sheaves, constructible sheaves and sheaves of modules over
quantizations. In each case, we discuss how "index formulas" allow us to easily
perform categorical calculations, and readily relate classical constructions of
geometric representation theory to categorical ones.Comment: 23 pages. an expository article to appear in "Perspectives on
Categorification.

### A categorical action on quantized quiver varieties

In this paper, we describe a categorical action of any Kac-Moody algebra on a
category of quantized coherent sheaves on Nakajima quiver varieties. By
"quantized coherent sheaves," we mean a category of sheaves of modules over a
deformation quantization of the natural symplectic structure on quiver
varieties. This action is a direct categorification of the geometric
construction of universal enveloping algebras by Nakajima.Comment: 26 pages. DVI may not compile correctly; PDF is recommended. v3:
extensive rewriting of proofs and exposition; main results are unchange

### Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all
representations of quantum groups. We show that these invariants coincide with
previous invariants defined by Khovanov for sl_2 and sl_3 and by
Mazorchuk-Stroppel and Sussan for sl_n.
Our technique is to study 2-representations of 2-quantum groups (in the sense
of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible
representations. These are the representation categories of certain finite
dimensional algebras with an explicit diagrammatic presentation, generalizing
the cyclotomic quotient of the KLR algebra. When the Lie algebra under
consideration is $\mathfrak{sl}_n$, we show that these categories agree with
certain subcategories of parabolic category O for gl_k.
We also investigate the finer structure of these categories: they are
standardly stratified and satisfy a double centralizer property with respect to
their self-dual modules. The standard modules of the stratification play an
important role as test objects for functors, as Vermas do in more classical
representation theory.
The existence of these representations has consequences for the structure of
previously studied categorifications. It allows us to prove the non-degeneracy
of Khovanov and Lauda's 2-category (that its Hom spaces have the expected
dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke
algebras are symmetric Frobenius.
In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps
between representations of quantum groups are used to define polynomial knot
invariants. We show that the categorifications of tensor products are related
by functors categorifying these maps, which allow the construction of bigraded
knot homologies whose graded Euler characteristics are the original polynomial
knot invariants.Comment: 99 pages. This is a significantly rewritten version of
arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been
significantly improved. These earlier papers have been left up mainly in the
interest of preserving references. v3: final version, to appear in Memoirs of
the AMS. Proof of nondegeneracy moved to separate erratu

### Rouquier's conjecture and diagrammatic algebra

We prove a conjecture of Rouquier relating the decomposition numbers in
category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's
canonical basis of a higher level Fock space. Independent proofs of this
conjecture have also recently been given by Rouquier, Shan, Varagnolo and
Vasserot and by Losev, using different methods.
Our approach is to develop two diagrammatic models for this category
$\mathcal{O}$; while inspired by geometry, these are purely diagrammatic
algebras, which we believe are of some intrinsic interest. In particular, we
can quite explicitly describe the representations of the Hecke algebra that are
hit by projectives under the $\mathsf{KZ}$-functor from the Cherednik category
$\mathcal{O}$ in this case, with an explicit basis.
This algebra has a number of beautiful structures including categorifications
of many aspects of Fock space. It can be understood quite explicitly using a
homogeneous cellular basis which generalizes such a basis given by Hu and
Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in
this diagrammatic formalism to category $\mathcal{O}$ for a cyclotomic rational
Cherednik algebra, including the connection of decomposition numbers to
canonical bases mentioned above, and an action of the affine braid group by
derived equivalences between different blocks.Comment: 64 pages; numerous TikZ figures, PDF is preferable to DVI. v4:
Revision in response to referee's report. Several proofs rewritten, examples
and pictures adde

### Cramped subgroups and generalized Harish-Chandra modules

Let G be a reductive complex Lie group with Lie algebra g. We call a subgroup
H of G {\bf cramped} if there is an integer b(G,H) such that each finite
dimensional representation of G has a non-trivial invariant subspace of
dimension less than b(G,H). We show that a subgroup is cramped if and only if
the moment map from T^*(K/L) to k^* is surjective, where K and L are compact
forms of G and H. We will use this in conjunction with sufficient conditions
for crampedness given by Willenbring and Zuckerman (2004) to prove a geometric
lemma on the intersections between adjoint orbits and Killing orthogonals to
subgroups. We will also discuss applications of the techniques of symplectic
geometry to the generalized Harish-Chandra modules introduced by Penkov and
Zuckerman (2004), of which our results on crampedness are special cases.Comment: 6 page

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