Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions

The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis \{h_\la\}_{\la\in\CY^k} to \{s_\la^{(k)}\}_{\la\in\CY^k} may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.Comment: 30 pages, 1 figur

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version of hep-th/0305038 which has been splitted in two articles. In this revised version, the introduction has been rewritten and a new appendix has been added. To appear in JP

Influence of a local change of depth on the behavior of bouncing oil drops

The work of Couder \textit{et al} (see also Bush \textit{et al}) inspired consideration of the impact of a submerged obstacle, providing a local change of depth, on the behavior of oil drops in the bouncing regime. In the linked videos, we recreate some of their results for a drop bouncing on a uniform depth bath of the same liquid undergoing vertical oscillations just below the conditions for Faraday instability, and show a range of new behaviors associated with change of depth. This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.Comment: High and low resolutions videos included as ancillary file

Tapered, tubular polyester fabric

A tapered tubular polyester sleeve is described to serve as the flexible foundation for a spacesuit limb covering. The tube has a large end and a small end with a length to be determined. The ratio of taper is also determined by scale factors. All the warp yarns extend to the large end. A requisite number of warp yarns extend the full length of the sleeve. Other warp yarns extend from the large end but are terminated along the length of the sleeve. It is then woven with a filling yarn which extends in a full circle along the full length of the sleeve to thereby define the tapered sleeve. The sleeve after fabrication is then placed on a mandrel, heated in an oven, and then attached to the arm or other limb of the spacesuit

Alien Registration- Lapointe, Charles J. (Jackman, Somerset County)

https://digitalmaine.com/alien_docs/7039/thumbnail.jp