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 Drinfel$'$d's new realization. Coproduct formulas are given and a PBW type basis is constructed.Comment: 15 page

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let $V$ be a $2m$-dimensional symplectic vector space over an algebraically closed field $K$. Let \mbb_n^{(f)} be the two-sided ideal of the Brauer algebra \mbb_n(-2m) over $K$ generated by $e_1e_3... e_{2f-1}$, where $0\leq f\leq [n/2]$. Let $\mathcal{HT}_{f}^{\otimes n}$ be the subspace of partially harmonic tensors of valence $f$ in $V^{\otimes n}$. In this paper, we prove that $\dim\mathcal{HT}_f^{\otimes n}$ and \dim\End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr) are both independent of $K$, and the natural homomorphism from \mbb_n(-2m)/\mbb_n^{(f)} to \End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr) is always surjective. We show that $\mathcal{HT}_{f}^{\otimes n}$ has a Weyl filtration and is isomorphic to the dual of V^{\otimes n}\mbb_n^{(f)}/V^{\otimes n}\mbb_n^{(f+1)} as a $Sp(V)$-(\mbb_n(-2m)/\mbb_n^{(f+1)})-bimodule. We obtain a $Sp(V)$-\mbb_n-bimodules filtration of $V^{\otimes n}$ such that each successive quotient is isomorphic to some \nabla(\lam)\otimes z_{g,\lam}\mbb_n with \lam\vdash n-2g, \ell(\lam)\leq m and $0\leq g\leq [n/2]$, where \nabla(\lam) is the co-Weyl module associated to \lam and z_{g,\lam} is an explicitly constructed maximal vector of weight \lam. As a byproduct, we show that each right \mbb_n-module z_{g,\lam}\mbb_n is integrally defined and stable under base change

On Automorphisms and Universal R-Matrices at Roots of Unity

Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity. There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations. One of these quotients, which is finite dimensional, has a universal R-matrix. In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other. We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case.Comment: 11 pages, minor TeXnical revision to allow automatic TeXin