On homological properties of strict polynomial functors of degree p

DOI10.1016/J.JALGEBRA.2022.11.015arXiv2106.05198MaRDI QIDQ6369895

Publication date: 9 June 2021

Abstract: We study the homological algebra in the category

m a t h c a l 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

m a t h c a l 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

m a t h c a l P_{p}

as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of

p

-weight

1

in

m a t h c a l P_{e}

for

e > p .

Mathematics Subject Classification ID

Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) (18G15) Derived categories, triangulated categories (18G80)

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