4,497 research outputs found
Diagrammatics for Coxeter groups and their braid groups
We give a monoidal presentation of Coxeter and braid 2-groups, in terms of
decorated planar graphs. This presentation extends the Coxeter presentation. We
deduce a simple criterion for a Coxeter group or braid group to act on a
category.Comment: Many figures, best viewed in color. Minor updates. This version
agrees with the published versio
Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements
Assume is a connected reductive algebraic group defined over an
algebraic closure of the finite field of
prime order . Furthermore, assume that is
a Frobenius endomorphism of . In this article we give a formula for
the value of any -stable character sheaf of at a unipotent
element. This formula is expressed in terms of class functions of
which are supported on a single unipotent class of .
In general these functions are not determined, however we give an expression
for these functions under the assumption that is connected,
is simple and is a good prime for .
In this case our formula is completely explicit.Comment: 29 pages. Parts of this article first appeared in arXiv:1306.5882.
This is an expanded and generalised of version of what appears there. (v2):
30 pages. Final version post referees report. Referenced work of
Digne-Lehrer-Michel who also independently obtained Theorem 7.
Microlocal approach to Lusztig's symmetries
We reformulate the De Concini -- Toledano Laredo conjecture about the
monodromy of the Casimir connection in terms of a relation between Lusztig's
symmetries of quantum group modules and the monodromy in the vanishing cycles
of factorizable sheaves.Comment: 20 pages, 1 figur
Moduli stacks of Serre stable representations in tilting theory
We introduce a new moduli stack, called the Serre stable moduli stack, which
corresponds to studying families of point objects in an abelian category with a
Serre functor. This allows us in particular, to re-interpret the classical
derived equivalence between most concealed-canonical algebras and weighted
projective lines by showing they are induced by the universal sheaf on the
Serre stable moduli stack. We explain why the method works by showing that the
Serre stable moduli stack is the tautological moduli problem that allows one to
recover certain nice stacks such as weighted projective lines from their moduli
of sheaves. As a result, this new stack should be of interest in both
representation theory and algebraic geometry
Partial mirror symmetry, lattice presentations and algebraic monoids
This is the second in a series of papers that develops the theory of
reflection monoids, motivated by the theory of reflection groups. Reflection
monoids were first introduced in arXiv:0812.2789. In this paper we study their
presentations as abstract monoids. Along the way we also find general
presentations for certain join-semilattices (as monoids under join) which we
interpret for two special classes of examples: the face lattices of convex
polytopes and the geometric lattices, particularly the intersection lattices of
hyperplane arrangements. Another spin-off is a general presentation for the
Renner monoid of an algebraic monoid, which we illustrate in the special case
of the "classical" algebraic monoids.Comment: 41 page
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