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

Generalised Stretched Littlewood-Richardson Coefficients

The Littlewood-Richardson (LR) coefficient counts among many other things the LR tableaux of a given shape and a given content. We prove, that the number of LR tableaux weakly increases if one adds to the shape and the content the shape and the content of another LR tableau. We also investigate the behaviour of the number of LR tableaux, if one repeatedly adds to the shape another shape with either fixed or arbitrary content. This is a generalisation of the stretched LR coefficients, where one repeatedly adds the same shape and content to itself.Comment: 15 pages, rewritten with more results and examples (compared with v1), final version to appear at Journal of Combinatorial Theory

Equality of multiplicity free skew characters

In this paper we show that two skew diagrams lambda/mu and alpha/beta can represent the same multiplicity free skew character [lambda/mu]=[alpha/beta] only in the the trivial cases when lambda/mu and alpha/beta are the same up to translation or rotation or if lambda=alpha is a staircase partition lambda=(l,l-1,...,2,1) and lambda/mu and alpha/beta are conjugate of each other.Comment: 16 pages, changes from v1 to v2: corrected the proof of Theorem 3.5 and some typos, changes from v2 to v3: minor layout change, enumeration changed, to appear in J. Algebraic Combi

On multiplicity-free skew characters and the Schubert Calculus

In this paper we classify the multiplicity-free skew characters of the symmetric group. Furthermore we show that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the `inverse' partitions (Theorem 4.2).Comment: 14 pages, to appear in Annals. Comb. minor changes from v1 to v2 as suggested by the referees, Example 3.4 inserted so numeration changed in section

Reduced Kronecker products which are multiplicity free or contain only few components

It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions \lambda,\mu are big enough then the coefficients of the Kronecker product [\lambda][\mu]=\sum_\n g(\l,\m,\n)[\nu] do not depend on the first part but only on the other parts. The reduced Kronecker product [\lambda]_\bullet \star[\mu]_\bullet can be viewed (roughly) as the Kronecker product [(n-|\lambda|,\lambda)][(n-|\mu|,\m)] for n big enough. In this paper we classify the reduced Kronecker products which are multiplicity free and those which contain less than 10 components.We furthermore give general lower bounds for the number of constituents and components of a given reduced Kronecker product. We also give a lower bound for the number of pairs of components whose corresponding partitions differ by one box. Finally we argue that equality of two reduced Kronecker products is only possible in the trivial case that the factors of the product are the same.Comment: 11 pages, final version. appears in European J. Combi

Schiefcharaktere und reduzierte Kronecker-Produkte von Charakteren der symmetrischen Gruppe

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On (almost) extreme components in Kronecker products of characters of the symmetric groups

Using a recursion formula due to Dvir, we obtain information on maximal and almost maximal components in Kronecker products of characters of the symmetric groups. This is applied to confirm a conjecture made by Bessenrodt and Kleshchev in 1999, which classifies all such Kronecker products with only three or four components.Comment: 38 pages, new theorems added, and paper now focused on maximal and almost maximal components in Kronecker product

Radiometric calibration of the in-flight blackbody calibration system of the GLORIA interferometer

GLORIA (Gimballed Limb Observer for Radiance Imaging of the Atmosphere) is an airborne, imaging, infrared Fourier transform spectrometer that applies the limb-imaging technique to perform trace gas and temperature measurements in the Earth's atmosphere with three-dimensional resolution. To ensure the traceability of these measurements to the International Temperature Scale and thereby to an absolute radiance scale, GLORIA carries an on-board calibration system. Basically, it consists of two identical large-area and high-emissivity infrared radiators, which can be continuously and independently operated at two adjustable temperatures in a range from −50 °C to 0 °C during flight. Here we describe the radiometric and thermometric characterization and calibration of the in-flight calibration system at the Reduced Background Calibration Facility of the Physikalisch-Technische Bundesanstalt. This was performed with a standard uncertainty of less than 110 mK. Extensive investigations of the system concerning its absolute radiation temperature and spectral radiance, its temperature homogeneity and its short- and long-term stability are discussed. The traceability chain of these measurements is presented

Characterization of blackbody inhomogeneity and its effect on the retrieval results of the GLORIA instrument

Limb sounding instruments play an important role in the monitoring of climate trends, as they provide a good vertical resolution. Traceability to the International System of Units (SI) via onboard reference or transfer standards is needed to compare trend estimates from multiple instruments. This study investigates the required uncertainty of these radiation standards to properly resolve decadal trends of climate-relevant trace species like ozone, water vapor, and temperature distribution for the Gimballed Limb Observer for Radiance Imaging of the Atmosphere (GLORIA). Temperature nonuniformities of the onboard reference blackbodies, used for radiometric calibration, have an impact on the calibration uncertainty. The propagation of these nonuniformities through the retrieval is analyzed. A threshold for the maximum tolerable uncertainty of the blackbody temperature is derived, so that climate trends can be significantly identified with GLORIA