Patryk Jaśniewski | Searcharxiv

arXiv2022

On homological properties of strict polynomial functors of degree p

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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arXiv2022

On strict polynomial functors with bounded domain

We introduce a new functor category: the category $\mathcal{P}_{d,n}$ of strict polynomial functors with bounded by $n$ domain of degree $d$ over a field of characteristic $p>0$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $\operatorname{GL}_n$. We investigate in detail the homological algebra in ${\cal P}_{d,n}$ for $d=p$ and establish equivalences between certain subcategories of ${\cal P}_{d,n}$'s which resemble the Spanier-Whitehead duality in stable homotopy theory.

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