Book Chapter10.1007/978-3-319-66065-3_9
Computing Weight Multiplicities
Pamela E. Harris
- 01 Jan 2017
- pp 193-222
4
TL;DR: Weyl alternation sets as discussed by the authors show interesting combinatorial and geometric properties, and can be used to investigate the number of terms contributing nontrivially to Kostant's weight multiplicity formula.
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Abstract: Central to the study of the representation theory of Lie algebras is the computation of weight multiplicities, which are the dimensions of vector subspaces called weight spaces. The multiplicity of a weight can be computed using a well-known formula of Kostant that consists of an alternating sum over a finite group and involves a partition function. In this paper, we specialize to the Lie algebra \(\mathfrak{s}\mathfrak{l}_{r+1}(\mathbb{C})\) and focus on questions regarding the number of terms contributing nontrivially to Kostant’s weight multiplicity formula. Through this study, we show that these contributing sets, called Weyl alternation sets, show interesting combinatorial and geometric properties. We dedicate a section to detailed examples that illustrate accessible techniques students may use to begin investigating the open problems we present in this area.
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Citations
Weight q-multiplicities for Representations of sp4(C)
TL;DR: In this paper, the q-analog of Kostant's partition function for the Lie algebra sp4(C) was used to give a simple formula for the qmultiplicity of a weight in the representations of the Lie algebras.
On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ sl 4 ( C )
Rebecca Garcia,Pamela E. Harris,Marissa Loving,Lucy Martinez,David Melendez,Joseph Rennie,Gordon Rojas Kirby,Daniel Tinoco +7 more
TL;DR: The q-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-Analog of kostant partition function as discussed by the authors, which gives the multiplicity of a weight in highest weight representation of a simple Lie algebra.
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On Kostant's weight $q$-multiplicity formula for $\mathfrak{sp}_6(\mathbb{C})$
TL;DR: In this article, a closed formula for the multiplicity of Kostant's partition function in the Weyl group was given for any pair of dominant integral weights of the Lie algebra (mathfrak{sp}_6(\mathbb{C})$.
1
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Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras
Pamela E. Harris,Marissa Loving,Juan Ramirez,Joseph Rennie,Gordon Rojas Kirby,Eduardo Torres Davila,Fabrice O. Ulysse +6 more
TL;DR: Weyl alternation sets as mentioned in this paper are subsets of the Weyl group which contribute nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so_4(C), so_5(C, sp_4 (C), and the exceptional Lie algebra g_2.
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