A geometric realization of Catalan functions
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917079650664448 |
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| author | Kato, Syu |
| author_facet | Kato, Syu |
| contents | We construct a smooth projective variety $\mathscr X_Ψ$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{GL}(n)$, determined by a root ideal $Ψ$. A natural family of line bundles on $\mathscr X_Ψ$ yields the Catalan functions -- symmetric functions introduced by Chen--Haiman and studied further by Blasiak--Morse--Pun--Summers. By analyzing the geometry of $\mathscr X_Ψ$, we prove the vanishing conjecture of Chen--Haiman, confirm the tame case of the vanishing conjecture of Blasiak--Morse--Pun, and establish the monotonicity conjectures of Shimozono--Weyman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00862 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A geometric realization of Catalan functions Kato, Syu Algebraic Geometry Combinatorics Representation Theory We construct a smooth projective variety $\mathscr X_Ψ$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{GL}(n)$, determined by a root ideal $Ψ$. A natural family of line bundles on $\mathscr X_Ψ$ yields the Catalan functions -- symmetric functions introduced by Chen--Haiman and studied further by Blasiak--Morse--Pun--Summers. By analyzing the geometry of $\mathscr X_Ψ$, we prove the vanishing conjecture of Chen--Haiman, confirm the tame case of the vanishing conjecture of Blasiak--Morse--Pun, and establish the monotonicity conjectures of Shimozono--Weyman. |
| title | A geometric realization of Catalan functions |
| topic | Algebraic Geometry Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2301.00862 |