Geodesic
When is the \(q\)-multiplicity of a weight a power of \(q\)?
Pamela E. Harris ; Margaret Rahmoeller1 ; Lisa Schneider2 ; Anthony Simpson3
1Roanoke College
2Salisbury University
3Williams College
The electronic journal of combinatorics, Tome 26 (2019) no. 4
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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 $q$.
DOI : 10.37236/8758
Classification : 05E10
Mots-clés : complex semisimple Lie algebra, irreducible representation, multiplicity, weight, primitive pair

Pamela E. Harris    ; Margaret Rahmoeller  1   ; Lisa Schneider  2   ; Anthony Simpson  3

1 Roanoke College
2 Salisbury University
3 Williams College 
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Pamela E. Harris; Margaret Rahmoeller; Lisa Schneider; Anthony Simpson. When is the \(q\)-multiplicity of a weight a power of \(q\)?. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/8758

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