Highest weight category structures on $rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916051271286784 |
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| author | Samokhin, Alexander van der Kallen, Wilberd |
| author_facet | Samokhin, Alexander van der Kallen, Wilberd |
| contents | Given a split simply connected and connected algebraic group scheme $\mathbb G$ over $\mathbb Z$ and a split parabolic subgroup scheme $\mathbb P\subset \mathbb G$, this paper constructs semi-orthogonal decompositions of the bounded derived category $D^b(\mathrm {rep}( \mathbb P))$ of noetherian representations of $\mathbb P$ with each semi-orthogonal component being equivalent to the bounded derived category $D^b(\mathrm {rep}( \mathbb G))$ of noetherian representations of $\mathbb G$. The semi-orthogonal components of those decompositions are stable under the monoidal action of $D^b(\mathrm {rep}( \mathbb G))$ on $D^b(\mathrm {rep}( \mathbb P))$. The decompositions depend on an arbitrarily chosen total order on the Weyl group that refines the Bruhat order. The semi-orthogonal decompositions are also compatible with the Bruhat order on cosets of the Weyl group of $\mathbb P$ in the Weyl group of $\mathbb G$. Their construction builds upon the foundational results on $\mathbb B$-modules from the works of Mathieu, Polo, and van der Kallen, and upon properties of the Steinberg basis of the $ \mathbb T$-equivariant $K$-theory of $ \mathbb G/\mathbb B$. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag schemes $\mathbb G/\mathbb P$ over $\mathbb Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13653 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Highest weight category structures on $rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$ Samokhin, Alexander van der Kallen, Wilberd Algebraic Geometry K-Theory and Homology Representation Theory 14F08, 14L30, 14M15, 19L47, 20G05, 20G10 Given a split simply connected and connected algebraic group scheme $\mathbb G$ over $\mathbb Z$ and a split parabolic subgroup scheme $\mathbb P\subset \mathbb G$, this paper constructs semi-orthogonal decompositions of the bounded derived category $D^b(\mathrm {rep}( \mathbb P))$ of noetherian representations of $\mathbb P$ with each semi-orthogonal component being equivalent to the bounded derived category $D^b(\mathrm {rep}( \mathbb G))$ of noetherian representations of $\mathbb G$. The semi-orthogonal components of those decompositions are stable under the monoidal action of $D^b(\mathrm {rep}( \mathbb G))$ on $D^b(\mathrm {rep}( \mathbb P))$. The decompositions depend on an arbitrarily chosen total order on the Weyl group that refines the Bruhat order. The semi-orthogonal decompositions are also compatible with the Bruhat order on cosets of the Weyl group of $\mathbb P$ in the Weyl group of $\mathbb G$. Their construction builds upon the foundational results on $\mathbb B$-modules from the works of Mathieu, Polo, and van der Kallen, and upon properties of the Steinberg basis of the $ \mathbb T$-equivariant $K$-theory of $ \mathbb G/\mathbb B$. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag schemes $\mathbb G/\mathbb P$ over $\mathbb Z$. |
| title | Highest weight category structures on $rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$ |
| topic | Algebraic Geometry K-Theory and Homology Representation Theory 14F08, 14L30, 14M15, 19L47, 20G05, 20G10 |
| url | https://arxiv.org/abs/2407.13653 |