The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order
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arXiv
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| Autori principali: | , , , , , , , , , |
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| Natura: | Preprint |
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2024
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| author | Anderson, Portia X. Banaian, Esther Ferreri, Melanie J. Goff, Owen C. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Mayers, Nicholas Wang, Shiyun Wilson, Alexander N. |
| author_facet | Anderson, Portia X. Banaian, Esther Ferreri, Melanie J. Goff, Owen C. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Mayers, Nicholas Wang, Shiyun Wilson, Alexander N. |
| contents | For integral weights $λ$ and $μ$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $μ$ in the irreducible representation with highest weight $λ$, which we denote by $m(λ,μ)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(λ,μ)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(λ,μ)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tildeα,μ)$, where $\tildeα$ is the highest root and $μ$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tildeα,μ)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $μ=-(α_i+α_{i+1}+\cdots+α_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16820 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order Anderson, Portia X. Banaian, Esther Ferreri, Melanie J. Goff, Owen C. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Mayers, Nicholas Wang, Shiyun Wilson, Alexander N. Representation Theory Combinatorics 05E10, 17B10, 17B22, 06A07 For integral weights $λ$ and $μ$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $μ$ in the irreducible representation with highest weight $λ$, which we denote by $m(λ,μ)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(λ,μ)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(λ,μ)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tildeα,μ)$, where $\tildeα$ is the highest root and $μ$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tildeα,μ)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $μ=-(α_i+α_{i+1}+\cdots+α_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. |
| title | The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order |
| topic | Representation Theory Combinatorics 05E10, 17B10, 17B22, 06A07 |
| url | https://arxiv.org/abs/2412.16820 |