Highest weight categories and strict polynomial functors. With an appendix by Cosima Aquilino

Krause, Henning

doi:10.4171/171-1/13

Cited by 12 publications

(11 citation statements)

References 27 publications

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“…The category of finitely generated modules over a quasi-hereditary algebra is an example of a highest weight category and conversely, every highest weight category with finitely many simple objects is of this form [22,Theorem 3.6]. Highest weight categories are also discussed in Krause's article in this volume [45]: in particular, the category of strict polynomial functors admits the structure of a highest weight category. Moreover, work of Dlab & Ringel [26] shows that every finite dimensional algebra admits a 'resolution' by a quasi-hereditary algebra.…”

Section: Introductionmentioning

confidence: 99%

Derived categories of quasi-hereditary algebras and their derived composition series

Kalck

2017

EMS Series of Congress Reports

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Abstract. We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confirms part of a conjecture of Bobiński & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composition factors. In other words, there is no Jordan-Hölder property for composition series of derived categories of quasi-hereditary algebras.

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Section: Introductionmentioning

confidence: 99%

Derived categories of quasi-hereditary algebras and their derived composition series

Kalck

2017

EMS Series of Congress Reports

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“…Proofs of these facts one can find in e.g. [1], [8], [10,11,14], [18]. In the second part of the section we introduce exact sequences, which will be crucial in the next sections.…”

Section: Preliminariesmentioning

confidence: 98%

On homological properties of strict polynomial functors of degree p

Jaśniewski¹

2021

Preprint

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We study the homological algebra in the category Pp 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 for simple functors and Schur functors and the Ext-groups between Schur and Weyl functors. We observe that the category Pp 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 Pp as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p-weight 1 in Pe for e > p.

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“…Our approach is motivated by that of [17], where Krause used the Cauchy decomposition of divided powers [1,12] to describe the highest weight structure of categories of strict polynomial functors. As Krause mentions, this leads to an alternate proof of the fact that classical Schur algebras S k (n, d) are quasi-hereditary, which follows by a Morita equivalence.…”

Section: Introductionmentioning

confidence: 99%