277 research outputs found
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
Box splines and the equivariant index theorem
In this article, we start to recall the inversion formula for the convolution
with the Box spline. The equivariant cohomology and the equivariant K-theory
with respect to a compact torus G of various spaces associated to a linear
action of G in a vector space M can be both described using some vector spaces
of distributions, on the dual of the group G or on the dual of its Lie algebra.
The morphism from K-theory to cohomology is analyzed and the multiplication by
the Todd class is shown to correspond to the operator (deconvolution) inverting
the semidiscrete convolution with a box spline. Finally, the multiplicities of
the index of a G-transversally elliptic operator on M are determined using the
infinitesimal index of the symbol.Comment: 44 page
Braid Group Action and Quantum Affine Algebras
We lift the lattice of translations in the extended affine Weyl group to a
braid group action on the quantum affine algebra. This action fixes the
Heisenberg subalgebra pointwise. Loop like generators are found for the algebra
which satisfy the relations of Drinfeld's new realization. Coproduct
formulas are given and a PBW type basis is constructed.Comment: 15 page
Polynomial Relations in the Centre of U_q(sl(N))
When the parameter of deformation q is a m-th root of unity, the centre of
U_q(sl(N))$ contains, besides the usual q-deformed Casimirs, a set of new
generators, which are basically the m-th powers of all the Cartan generators of
U_q(sl(N)). All these central elements are however not independent. In this
letter, generalising the well-known case of U_q(sl(2)), we explicitly write
polynomial relations satisfied by the generators of the centre. Application to
the parametrization of irreducible representations and to fusion rules are
sketched.Comment: 8 pages, minor TeXnical revision to allow automatic TeXin
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