Minkowski difference weight formulas
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912035090989056 |
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| author | Teja, G. Krishna |
| author_facet | Teja, G. Krishna |
| contents | Fix any complex Kac-Moody Lie algebra $\mathfrak{g}$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. We study arbitrary highest weight $\mathfrak{g}$-modules $V$ (with any highest weight $λ\in \mathfrak{h}^*$, and let $L(λ)$ be the corresponding simple highest weight $\mathfrak{g}$-module), and write their weight-sets $\mathrm{wt} V$. This is based on and generalizes the Minkowski decompositions for all $\mathrm{wt} L(λ)$ and hulls $\mathrm{conv}_{\mathbb{R}}(\mathrm{wt} V)$, of Khare [J. Algebra. 2016 & Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 & J. Algebra. 2022]. Those works need a freeness property of the Dynkin graph nodes of integrability $J_λ$ of $L(λ)$: $\mathrm{wt} L(λ)\ -$ any sum of simple roots over $J_λ^c$ are all weights of $L(λ)$. We generalize it for all $V$, by introducing nodes $J_V$ that record all the lost 1-dim. weights in $V$. We show three applications (seemingly novel) for all $\big(\mathfrak{g}, λ, V\big)$ of our $J_V^c$-freeness: 1) Minkowski decompositions of all $\mathrm{wt} V$, subsuming those above for simples. 1$'$) Characterization of these formulas. 1$''$) For these, we solve the inverse problem of determining all $V$ with fixing $\mathrm{wt} V \ =$ weight-set of a Verma, parabolic Verma and $L(λ)$ $\forall$ $λ$. 2) At module level (by raising operators' actions), construction of weight vectors along $J_V^c$-directions. 3) Lower bounds on the multiplicities of such weights, in all $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minkowski difference weight formulas Teja, G. Krishna Representation Theory Primary: 17B10, Secondary: 17B20, 17B22, 17B67, 17B70, 52B20, 52B99 Fix any complex Kac-Moody Lie algebra $\mathfrak{g}$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. We study arbitrary highest weight $\mathfrak{g}$-modules $V$ (with any highest weight $λ\in \mathfrak{h}^*$, and let $L(λ)$ be the corresponding simple highest weight $\mathfrak{g}$-module), and write their weight-sets $\mathrm{wt} V$. This is based on and generalizes the Minkowski decompositions for all $\mathrm{wt} L(λ)$ and hulls $\mathrm{conv}_{\mathbb{R}}(\mathrm{wt} V)$, of Khare [J. Algebra. 2016 & Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 & J. Algebra. 2022]. Those works need a freeness property of the Dynkin graph nodes of integrability $J_λ$ of $L(λ)$: $\mathrm{wt} L(λ)\ -$ any sum of simple roots over $J_λ^c$ are all weights of $L(λ)$. We generalize it for all $V$, by introducing nodes $J_V$ that record all the lost 1-dim. weights in $V$. We show three applications (seemingly novel) for all $\big(\mathfrak{g}, λ, V\big)$ of our $J_V^c$-freeness: 1) Minkowski decompositions of all $\mathrm{wt} V$, subsuming those above for simples. 1$'$) Characterization of these formulas. 1$''$) For these, we solve the inverse problem of determining all $V$ with fixing $\mathrm{wt} V \ =$ weight-set of a Verma, parabolic Verma and $L(λ)$ $\forall$ $λ$. 2) At module level (by raising operators' actions), construction of weight vectors along $J_V^c$-directions. 3) Lower bounds on the multiplicities of such weights, in all $V$. |
| title | Minkowski difference weight formulas |
| topic | Representation Theory Primary: 17B10, Secondary: 17B20, 17B22, 17B67, 17B70, 52B20, 52B99 |
| url | https://arxiv.org/abs/2409.12802 |