Spherical functions and aq-analogue of Kostant's weight multiplicity formula

1982 ◽  
Vol 66 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Shin-ichi Kato
Keyword(s):  
Spherical Functions ◽  
Weight Multiplicity ◽  
Multiplicity Formula

scholarly journals When is the $q$-Multiplicity of a Weight a Power of $q$?

10.37236/8758 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Pamela E. Harris ◽  
Margaret Rahmoeller ◽  
Lisa Schneider ◽  
Anthony Simpson
Keyword(s):  
Lie Algebras ◽  
Dynkin Diagram ◽  
Fibonacci Numbers ◽  
Simple Lie Algebras ◽  
Weight Multiplicity ◽  
Highest Weight ◽  
Multiplicity Formula ◽  
Exhaustive List

Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights $(\lambda,\mu)$ of simple Lie algebras $\mathfrak{g}$ (up to Dynkin diagram isomorphism) for which the multiplicity of the weight $\mu$ in the representation of $\mathfrak{g}$ with highest weight $\lambda$ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated $q$-multiplicity for the pairs of weights considered, and conjecture that in all cases the $q$-multiplicity of such pairs of weights is given by a power of $q$.

Spheric radial basis functions in Mahalanobis measuring for displaying local field features

2019 ◽  
Vol 952 (10) ◽  
pp. 2-9
Author(s):  
Yu.M. Neiman ◽  
L.S. Sugaipova ◽  
V.V. Popadyev
Keyword(s):  
Gravitational Field ◽  
Quadratic Form ◽  
Local Field ◽  
Euclidean Distance ◽  
Basis Functions ◽  
Spherical Functions ◽  
Local Models ◽  
Stationary Field ◽  
Modeling Process ◽  
The Earth

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

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