Multiplicities and tensor product coefficients for ArA_r

We apply some recent developments of Baldoni-DeLoera-Vergne on vector partition functions, to Kostant and Steinberg formulas, in the case of $A_r$. We therefore get a fast {\sc Maple} program that computes for $A_r$: the multiplicity $c_{\lambda,\mu}$ of the weight $\mu$ in the representation $V(\lambda)$ of highest weight $\lambda$; the multiplicity $c_{\lambda,\mu,\nu}$ of the representation $V(\nu)$ in $V(\lambda)\otimes V(\mu)$. The computation also gives the locally polynomial functions $c_{\lambda,\mu}$ and $c_{\lambda,\mu,\nu}$

Chopped and sliced cones and representations of Kac-Moody algebras

We introduce the notion of a chopped and sliced cone in combinatorial geometry and prove two structure theorems for the number of integral points in the individual slices of such a cone. We observe that this notion applies to weight multiplicities of Kac-Moody algebras and Littlewood-Richardson coefficients of semisimple Lie algebras, where we obtain the corresponding results.Comment: 9 pages, 1 figur

Graded Betti numbers of powers of ideals

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive \ZZ-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of \ZZ-grading, \ZZ^2 can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu, t)$, \dim_k \left(\tor_i^S(I^t, k)_{\mu} \right) is equal to one of these polynomials in $(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a \ZZ^d-graded algebra, for a positive grading.Comment: 20 page

Vertices of Gelfand-Tsetlin Polytopes

This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory \mathfrak{gl}_n \C and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.Comment: 14 pages, 3 figures, fixed attribution