On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929261828374528 |
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| author | Boysal, Arzu |
| author_facet | Boysal, Arzu |
| contents | For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $ρ$ be the half sum of positive roots and $P(2ρ)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $λ$ of the form $λ=2ρ-\sum_{i=1}^r a_iα_i$ with $a_i\in \mathbb{Z}_{\geq 0}$ for $1\leq i\leq r$.
We show that if $λ$ is a vertex of $P(2ρ)$, then $V(λ)$ appears in $V(ρ) \otimes V(ρ)$ with multiplicity one, proving partially (for the vertices of $P(2ρ)$) a conjecture of Kostant describing components of $V(ρ)\otimes V(ρ)$. This result allows us to give an alternative proof for a weaker form of the conjecture (up to saturation factor) for any $\mathfrak{g}$. Further, using works of Knutson-Tau on the saturation property of $\mathfrak{sl_{r+1}}$, our results give an alternative proof of Kostant's conjecture in the particular case $\mathfrak{g}=\mathfrak{sl_{r+1}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06890 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$ Boysal, Arzu Representation Theory 17B10 For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $ρ$ be the half sum of positive roots and $P(2ρ)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $λ$ of the form $λ=2ρ-\sum_{i=1}^r a_iα_i$ with $a_i\in \mathbb{Z}_{\geq 0}$ for $1\leq i\leq r$. We show that if $λ$ is a vertex of $P(2ρ)$, then $V(λ)$ appears in $V(ρ) \otimes V(ρ)$ with multiplicity one, proving partially (for the vertices of $P(2ρ)$) a conjecture of Kostant describing components of $V(ρ)\otimes V(ρ)$. This result allows us to give an alternative proof for a weaker form of the conjecture (up to saturation factor) for any $\mathfrak{g}$. Further, using works of Knutson-Tau on the saturation property of $\mathfrak{sl_{r+1}}$, our results give an alternative proof of Kostant's conjecture in the particular case $\mathfrak{g}=\mathfrak{sl_{r+1}}$. |
| title | On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$ |
| topic | Representation Theory 17B10 |
| url | https://arxiv.org/abs/2309.06890 |