Etienne Rassart

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We present a polynomiality property of the Littlewood-Richardson coefficients c λμ . The coefficients are shown to be given by polynomials in λ, μ and ν on the cones of the chamber complex of a vector partition function. We give bounds on the degree of the polynomials depending on the maximum allowed number of parts of the partitions λ, μ and ν. We first(More)
We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the β-Hermite and β-Laguerre ensembles of(More)
We use Gelfand–Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra slkC (type Ak−1) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat–Heckman measure from symplectic(More)
Using tools from combinatorics, convex geometry and symplectic geometry, we study the behavior of the Kostka numbers Kλβ and Littlewood-Richardson coefficients cλμ (the type A weight multiplicities and Clebsch-Gordan coefficients). We show that both are given by piecewise polynomial functions in the entries of the partitions and compositions parametrizing(More)
We discuss some applications of signature quantization to the representation theory of compact Lie groups. In particular, we prove signature analogues of the Kostant formula for weight multiplicities and the Steinberg formula for tensor product multiplicities. Using symmetric functions, we also find, for type A, analogues of the Weyl branching rule and the(More)
Parallelogram polyominoes are a subclass of convex polyominoes in the square lattice that has been studied extensively in the literature. Recently [18] congruence classes of convex polyominoes with respect to rotations and reflections have been enumerated by counting orbits under the action of the dihedral group D4, of symmetries of the square, on(More)
We discuss some applications of signature quantization to the representation theory of compact Lie groups. In particular, we prove signature analogues of the Kostant formula for weight multiplicities and the Steinberg formula for tensor product multiplicities. Using symmetric functions, we also find, for type A, analogues of the Weyl branching rule and the(More)
We discuss a family of representations of Lie groups related to quantization with respect to the Dirac signature operator. The combinatorics of these twisted representations is similar to that of the usual irreducible representations, but involve a specialization of a q-analogue of the Kostant partition function. In particular, we prove signature analogues(More)
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