625 research outputs found

### Wonderful models for toric arrangements

We build a wonderful model for toric arrangements. We develop the "toric
analog" of the combinatorics of nested sets, which allows to define a family of
smooth open sets covering the model. In this way we prove that the model is
smooth, and we give a precise geometric and combinatorial description of the
normal crossing divisor.Comment: Final version, to appear on IMRN. 23 pages, 1 pictur

### Simple linear compactifications of odd orthogonal groups

We classify the simple linear compactifications of SO(2r+1), namely those
compactifications with a unique closed orbit which are obtained by taking the
closure of the SO(2r+1)xSO(2r+1)-orbit of the identity in a projective space
P(End(V)), where V is a finite dimensional rational SO(2r+1)-module.Comment: v2: several simplifications, final version. To appear in J. Algebr

### Topological invariants from non-restricted quantum groups

We introduce the notion of a relative spherical category. We prove that such
a category gives rise to the generalized Kashaev and Turaev-Viro-type
3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624,
respectively. In this case we show that these invariants are equal and extend
to what we call a relative Homotopy Quantum Field Theory which is a branch of
the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our
main examples of relative spherical categories are the categories of finite
dimensional weight modules over non-restricted quantum groups considered by C.
De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories
are not semi-simple and have an infinite number of non-isomorphic irreducible
modules all having vanishing quantum dimensions. We also show that these
categories have associated ribbon categories which gives rise to re-normalized
link invariants. In the case of sl(2) these link invariants are the
Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and
T. Ohtsuki.Comment: 37 pages, 16 figure

### On some modules of covariants for a reflection group

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak
h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes
\bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes
\mathfrak g)^\mathfrak g$ of
$(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules, where
$\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced
form of a conjecture of Reeder for the adjoint representation.
New version with different title. Various improvements. New section 7.Comment: 18 Page

### The algebra of the box spline

In this paper we want to revisit results of Dahmen and Micchelli on
box-splines which we reinterpret and make more precise. We compare these ideas
with the work of Brion, Szenes, Vergne and others on polytopes and partition
functions.Comment: 69 page

### Nesting maps of Grassmannians

Let F be a field and i < j be integers between 1 and n. A map of
Grassmannians f : Gr(i, F^n) --> Gr(j, F^n) is called nesting, if l is
contained in f(l) for every l in Gr(i, F^n). We show that there are no
continuous nesting maps over C and no algebraic nesting maps over any
algebraically closed field F, except for a few obvious ones. The continuous
case is due to Stong and Grover-Homer-Stong; the algebraic case in
characteristic zero can also be deduced from their results. In this paper we
give new proofs that work in arbitrary characteristic. As a corollary, we give
a description of the algebraic subbundles of the tangent bundle to the
projective space P^n over F. Another application can be found in a recent paper
math.AC/0306126 of George Bergman

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