On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$

[2309.06890] On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$ Skip to main content Search Submit Donate Log in Mathematics > Representation Theory arXiv:2309.06890 (math) [Submitted on 13 Sep 2023 ( v1 ), last revised 3 Mar 2024 (this version, v2)] Title: On Kostant's conjecture for components of $V(ρ)\otimes V(ρ)$ Authors: Arzu Boysal View PDF HTML (experimental) Abstract: For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $\rho$ be the half sum of positive roots and $P(2\rho)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $\lambda$ of the form $\lambda=2\rho-\sum_{i=1}^r a_i\alpha_i$ with $a_i\in \mathbb{Z}_{\geq 0}$ for $1\leq i\leq r$. We show that if $\lambda$ is a vertex of $P(2\rho)$, then $V(\lambda)$ appears in $V(\rho) \otimes V(\rho)$ with multiplicity one, proving partially (for the vertices of $P(2\rho)$) a conjecture of Kostant describing components of $V(\rho)\otimes V(\rho)$. 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 $\mathfrak{g}=\mathfrak{sl_{r+1}}$. Comments: 12 pages, a new section (2.2) on vertices of dominant weight polytopes added, following sections are re-organized with additional comments Subjects: Representation Theory (math.RT) MSC classes: 17B10 Cite as: arXiv:2309.06890 [math.RT] (or arXiv:2309.06890v2 [math.RT] for this version) https://doi.org/10.48550/arXiv.2309.06890 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s10468-026-10410-8 Focus to learn more DOI(s) linking to related resources Submission history From: Arzu Boysal [ view email ] [v1] Wed, 13 Sep 2023 11:27:20 UTC (10 KB) [v2] Sun, 3 Mar 2024 13:07:23 UTC (11 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2023-09 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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