16,700 research outputs found
On the number of minimal partitions of a box into boxes
AbstractThe number of minimal partitions of a box into proper boxes is examined
On principal hook length partitions and durfee sizes in skew characters
In this paper we construct for a given arbitrary skew diagram A all
partitions nu with maximal principal hook lengths among all partitions with the
character [nu] appearing in the skew character [A]. Furthermore we show that
these are also partitions with minimal Durfee size. This we use to give the
maximal Durfee size for [nu] appearing in [A] for the cases when A decays into
two partitions and for some special cases of A. Also this gives conditions for
two skew diagrams to represent the same skew character.Comment: 13 pages, minor changes from v1 to v2 as suggested by the referee, to
appear in Annals. Com
Abacus models for parabolic quotients of affine Weyl groups
We introduce abacus diagrams that describe minimal length coset
representatives in affine Weyl groups of types B, C, and D. These abacus
diagrams use a realization of the affine Weyl group of type C due to Eriksson
to generalize a construction of James for the symmetric group. We also describe
several combinatorial models for these parabolic quotients that generalize
classical results in affine type A related to core partitions.Comment: 28 pages, To appear, Journal of Algebra. Version 2: Updated with
referee's comment
Gluing two affine Yangians of
We construct a four-parameter family of affine Yangian algebras by gluing two
copies of the affine Yangian of . Our construction allows for
gluing operators with arbitrary (integer or half integer) conformal dimension
and arbitrary (bosonic or fermionic) statistics, which is related to the
relative framing. The resulting family of algebras is a two-parameter
generalization of the affine Yangian, which is isomorphic to
the universal enveloping algebra of . All algebras that we construct
have natural representations in terms of "twin plane partitions", a pair of
plane partitions appropriately joined along one common leg. We observe that the
geometry of twin plane partitions, which determines the algebra, bears striking
similarities to the geometry of certain toric Calabi-Yau threefolds.Comment: 88 pages, 12 figure
The blocks of the Brauer algebra in characteristic zero
We determine the blocks of the Brauer algebra in characteristic zero. We also give information on the submodule structure of standard modules for this algebra
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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