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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2312.09986 |
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| _version_ | 1866916357707137024 |
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| author | Harry, Kimberly J. |
| author_facet | Harry, Kimberly J. |
| contents | Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $μ$ in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$, which we denote $L(\tildeα)$, where $\tildeα$ is the highest root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $μ=α_i+α_{i+1}+\cdots+α_j$ with $1\leq i\leq j\leq r$ in $L(\tildeα)$ is given by the product $F_{i}\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $μ=α_i+α_{i+1}+\cdots+α_j$ with $1\leq i\leq j\leq r$ in the representation $L(\tildeα)$ is precisely $q^{r-h(μ)}$, where $h(μ)=j-i+1$ is the height of the positive root $μ$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$ is one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09986 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computing the $q$-Multiplicity of the Positive Roots of $\mathfrak{sl}_{r+1}(\mathbb{C})$ and Products of Fibonacci Numbers Harry, Kimberly J. Representation Theory Combinatorics 05E10 Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $μ$ in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$, which we denote $L(\tildeα)$, where $\tildeα$ is the highest root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $μ=α_i+α_{i+1}+\cdots+α_j$ with $1\leq i\leq j\leq r$ in $L(\tildeα)$ is given by the product $F_{i}\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $μ=α_i+α_{i+1}+\cdots+α_j$ with $1\leq i\leq j\leq r$ in the representation $L(\tildeα)$ is precisely $q^{r-h(μ)}$, where $h(μ)=j-i+1$ is the height of the positive root $μ$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$ is one. |
| title | Computing the $q$-Multiplicity of the Positive Roots of $\mathfrak{sl}_{r+1}(\mathbb{C})$ and Products of Fibonacci Numbers |
| topic | Representation Theory Combinatorics 05E10 |
| url | https://arxiv.org/abs/2312.09986 |