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arXiv · 2106.05198

On homological properties of strict polynomial functors of degree p

Patryk Jaśniewski
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Abstract

We study the homological algebra in the category $\mathcal{P}_p$ of strict polynomial functors of degree $p$ over a field of positive characteristic $p$. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category $\mathcal{P}_p$ has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of $\mathcal{P}_p$ as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of $p$-weight $1$ in $\mathcal{P}_e$ for $e>p.$

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Patryk Jaśniewski. 2021-06-09. On homological properties of strict polynomial functors of degree p. https://arxiv.org/abs/2106.05198

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