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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 present a polynomiality property of the Littlewood-Richardson coefficients c ν 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 use Gelfand–Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl k C (type A k−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)

- Etienne Rassart, Sara Billey, François Blanchette, Peter Clifford, Ioana Dumitriu, Alex Ghitza +7 others
- 2004

Using tools from combinatorics, convex geometry and symplectic geometry, we study the behavior of the Kostka numbers K λβ and Littlewood-Richardson coefficients c ν λµ (the type A weight multi-plicities and Clebsch-Gordan coefficients). We show that both are given by piecewise polynomial functions in the entries of the partitions and compositions… (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)

- Victor Guillemin, Etienne Rassart
- 2004

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)

- Victor Guillemin, Etienne Rassart
- 2004

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)

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