979 research outputs found
Inequalities between Littlewood–Richardson coefficients
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to
ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.Natural Sciences and Engineering Research Council of CanadaFonds Québécois de la Recherche sur la Nature et les Technologie
Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in
\cite{Chi08}, we use quiver theory to represent the generalized
Littlewood-Richardson coefficients for the branching rule for the diagonal
embedding of \gl(n) as the dimension of a weight space of semi-invariants.
Using this, we prove their saturation and investigate when they are nonzero. We
also show that for certain partitions the associated stretched polynomials
satisfy the same conjectures as single Littlewood-Richardson coefficients. We
then provide a polytopal description of this multiplicity and show that its
positivity may be computed in strongly polynomial time. Finally, we remark that
similar results hold for certain other generalized Littlewood-Richardson
coefficients.Comment: 28 pages, comments welcom
On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect
We investigate the problem of computing tensor product multiplicities for
complex semisimple Lie algebras. Even though computing these numbers is #P-hard
in general, we show that if the rank of the Lie algebra is assumed fixed, then
there is a polynomial time algorithm, based on counting the lattice points in
polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based
on the ellipsoid algorithm, to decide when the coefficients are nonzero in
polynomial time for arbitrary rank. Our experiments show that the lattice point
algorithm is superior in practice to the standard techniques for computing
multiplicities when the weights have large entries but small rank. Using an
implementation of this algorithm, we provide experimental evidence for
conjectured generalizations of the saturation property of
Littlewood--Richardson coefficients. One of these conjectures seems to be valid
for types B_n, C_n, and D_n.Comment: 21 pages, 6 table
A product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials
Jack polynomials generalize several classical families of symmetric
polynomials, including Schur polynomials, and are further generalized by
Macdonald polynomials. In 1989, Richard Stanley conjectured that if the
Littlewood-Richardson coefficient for a triple of Schur polynomials is 1, then
the corresponding coefficient for Jack polynomials can be expressed as a
product of weighted hooks of the Young diagrams associated to the partitions
indexing the coefficient. We prove a special case of this conjecture in which
the partitions indexing the Littlewood-Richardson coefficient have at most 3
parts. We also show that this result extends to Macdonald polynomials.Comment: 30 page
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