City Research Online

Abstract

This thesis focuses on perverse autoequivalences of finite-dimensional symmetric algebras A over an algebraically closed field k. These are autoequivalences, introduced by Chuang and Rouquier, of the bounded derived category Dᵇ(A) of A-modules filtered by shifted Morita equivalences. We pay special attention to what we call two-step perverse autoequivalences, for which the filtration is of length two.
In particular, we demonstrate that two-step perverse autoequivalences of a certain form give rise to distinguished modules in the endomorphism algebra E of some projective A-module exhibiting a property we term strong periodicity. Such modules are periodic, and the periodicity arises from an extension of the E-E-bimodule E by itself.
We then prove a converse: given strongly periodic E-modules, we construct endofunctors of Dᵇ(A), and prove that these endofunctors are two-step perverse autoequivalences. This is closely related to work of Grant on perverse autoequivalences arising from periodic endomorphism algebras, and our result encompasses his. Building on Grant's work, we show that these autoequivalences coincide with iterated combinatorial tilts, as defined by Rickard and Okuyama.
Finally, we survey some applications of our result to blocks of symmetric groups of weight two. This recovers equivalences of Craven and Rouquier, arising from the geometry of the underlying groups, while our methods are based only on the algebras themselves.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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