Canonical Landau-Ginzburg models for cominuscule homogeneous spaces
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910637506953216 |
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| author | Spacek, Peter Wang, Charles |
| author_facet | Spacek, Peter Wang, Charles |
| contents | We present a type-independent Landau-Ginzburg (LG) model $(X_\mathrm{can}, \mathcal{W}_\mathrm{can})$ for any cominuscule homogeneous space $X=G/P$. We give a fully combinatorial construction for our superpotential $\mathcal{W}_\mathrm{can}$ as a sum of $n+1$ rational functions in the (generalized) Plücker coordinates on the "Langlands dual" minuscule homogeneous space $\mathbb{X}=P^\vee\backslash G^\vee$. Explicitly, we define the denominators $\mathcal{D}_{i_*}$ of these rational functions using the combinatorics of order ideals of the corresponding minuscule poset, which can be interpreted as (generalized) Young diagrams, by a process that can be described by "moving boxes" and hence is easily implemented. To construct the corresponding numerators, we define derivations $δ_{i_1}$ on $\mathbb{C}[\mathbb{X}]$ that act by "adding an appropriate box if possible" and then we apply each $δ_{i_1}$ to the corresponding $\mathcal{D}_{i_*}$. By studying certain Weyl orbits in the fundamental representations of $\widetilde{G}^\vee$ and exploiting the existence of a certain dense algebraic torus in $\mathbb{X}$, we show that the polynomials $\mathcal{D}_{i_*}$ coincide with the generalized minors $ϕ_{i_*}$ appearing in the cluster structures for homogeneous spaces studied by Geiß-Leclerc-Schröer in arXiv:math/0609138. We then define the mirror variety $X_\mathrm{can}=\mathbb{X}\setminus D_\mathrm{ac}$ to be the complement of the anticanonical divisor $D_\mathrm{ac} = \sum_{i_*}\{\mathcal{D}_{i_*}=0\}$ formed by the $\mathcal{D}_{i_*}$. Moreover, we show that the LG models $(X_\mathrm{can},\mathcal{W}_\mathrm{can})$ are isomorphic to the Lie-theoretic LG-models $(X_\mathrm{Lie},\mathcal{W}_\mathrm{Lie})$ constructed by Rietsch in arXiv:math/0511124 and our models naturally generalize the type-dependent Plücker coordinate LG-models previously studied by various authors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05070 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canonical Landau-Ginzburg models for cominuscule homogeneous spaces Spacek, Peter Wang, Charles Algebraic Geometry Combinatorics Representation Theory 14J33, 14M17, 14N35, 05E14, 05E10, 20G20 We present a type-independent Landau-Ginzburg (LG) model $(X_\mathrm{can}, \mathcal{W}_\mathrm{can})$ for any cominuscule homogeneous space $X=G/P$. We give a fully combinatorial construction for our superpotential $\mathcal{W}_\mathrm{can}$ as a sum of $n+1$ rational functions in the (generalized) Plücker coordinates on the "Langlands dual" minuscule homogeneous space $\mathbb{X}=P^\vee\backslash G^\vee$. Explicitly, we define the denominators $\mathcal{D}_{i_*}$ of these rational functions using the combinatorics of order ideals of the corresponding minuscule poset, which can be interpreted as (generalized) Young diagrams, by a process that can be described by "moving boxes" and hence is easily implemented. To construct the corresponding numerators, we define derivations $δ_{i_1}$ on $\mathbb{C}[\mathbb{X}]$ that act by "adding an appropriate box if possible" and then we apply each $δ_{i_1}$ to the corresponding $\mathcal{D}_{i_*}$. By studying certain Weyl orbits in the fundamental representations of $\widetilde{G}^\vee$ and exploiting the existence of a certain dense algebraic torus in $\mathbb{X}$, we show that the polynomials $\mathcal{D}_{i_*}$ coincide with the generalized minors $ϕ_{i_*}$ appearing in the cluster structures for homogeneous spaces studied by Geiß-Leclerc-Schröer in arXiv:math/0609138. We then define the mirror variety $X_\mathrm{can}=\mathbb{X}\setminus D_\mathrm{ac}$ to be the complement of the anticanonical divisor $D_\mathrm{ac} = \sum_{i_*}\{\mathcal{D}_{i_*}=0\}$ formed by the $\mathcal{D}_{i_*}$. Moreover, we show that the LG models $(X_\mathrm{can},\mathcal{W}_\mathrm{can})$ are isomorphic to the Lie-theoretic LG-models $(X_\mathrm{Lie},\mathcal{W}_\mathrm{Lie})$ constructed by Rietsch in arXiv:math/0511124 and our models naturally generalize the type-dependent Plücker coordinate LG-models previously studied by various authors. |
| title | Canonical Landau-Ginzburg models for cominuscule homogeneous spaces |
| topic | Algebraic Geometry Combinatorics Representation Theory 14J33, 14M17, 14N35, 05E14, 05E10, 20G20 |
| url | https://arxiv.org/abs/2410.05070 |