Periodic Reporting for period 1 - COMBGEOREP (Combinatorial and geometric methods in representation theory)
Okres sprawozdawczy: 2020-01-20 do 2022-01-19
The overall goal is to develop homological and geometric methods to study representations of algebras, by 1) using the geometry of Riemann surfaces to study skewed-gentle algebras and their tau-tilting theory, and 2) developing cluster theoretic techniques in negative Calabi-Yau (CY) triangulated categories.
Joint work with K. Baur was carried out towards the construction of a geometric model for skewed-gentle algebras in the first three months of the grant. However, in the meantime very similar results by other authors appeared on the arXiv (arXiv:2004.11136). We thus considered the class of string algebras instead, and obtained a geometric model for their module categories via partial triangulations of punctured surfaces, and a combinatorial description of their tau-tilting theory.
I started a new project with Emily Barnard, Emily Gunawan and Ralf Schiffler on a new class of representation theoretic objects, that of maximal almost rigid objects, By considering the geometric model constructed by K. Baur and I, we have a combinatorial description of maximal almost rigid objects over gentle algebras in terms of so-called permissible triangulations of a surface. This project is current work in progress.
In joint work with D. Pauksztello and D. Ploog, we established a bijection between simple-minded collections in the bounded derived category of an hereditary algebra sitting in the fundamental domain of the negative CY cluster category C and simpleminded systems in C. We also obtained a description of functorially finite hearts, which was key to obtain the bijection above.
Furthermore, we obtained a relationship between simple-minded systems and positive noncrossing partitions, generalising previous results by Iyama-Jin, Buan-Reiten-
Thomas and by me. These results are now published in Compositio Mathematica. This manuscript contains also an appendix containing joint work with D. Pauksztello
and A. Zvonareva on a reduction technique for simple-minded collections which does not involve Verdier localisations.
Partial results were also obtained on the mutation behaviour of simple-minded objects. In joint work with David Pauksztello, we showed that the left and right
mutations of w-simple-minded systems are again w-simple-minded systems, and that an almost complete w-simple-minded system has at least w complements. We also established an analogue result for simple-minded collections for which the extension closure is functorially finite. Moreover, we have a compatibility result between these mutations and HRS-tilting.
I started a new project with Emily Barnard, Emily Gunawan and Ralf Schiffler on a new class of representation theoretic objects, that of maximal almost rigid objects, By considering the geometric model constructed by K. Baur and I, we have a combinatorial description of maximal almost rigid objects over gentle algebras in terms of so-called permissible triangulations of a surface. This project is current work in progress.
In joint work with D. Pauksztello and D. Ploog, we established a bijection between simple-minded collections in the bounded derived category of an hereditary algebra sitting in the fundamental domain of the negative CY cluster category C and simpleminded systems in C. We also obtained a description of functorially finite hearts, which was key to obtain the bijection above.
Furthermore, we obtained a relationship between simple-minded systems and positive noncrossing partitions, generalising previous results by Iyama-Jin, Buan-Reiten-
Thomas and by me. These results are now published in Compositio Mathematica. This manuscript contains also an appendix containing joint work with D. Pauksztello
and A. Zvonareva on a reduction technique for simple-minded collections which does not involve Verdier localisations.
Partial results were also obtained on the mutation behaviour of simple-minded objects. In joint work with David Pauksztello, we showed that the left and right
mutations of w-simple-minded systems are again w-simple-minded systems, and that an almost complete w-simple-minded system has at least w complements. We also established an analogue result for simple-minded collections for which the extension closure is functorially finite. Moreover, we have a compatibility result between these mutations and HRS-tilting.
The joint published work with D. Pauksztello and D. Ploog gives a generalisation of previous results in the literature, for which new tools and techniques were needed. Namely, the result on the funtorially finite hearts holds independent interest and is likely to be widely applicable in homological algebra and representation theory, as it has been already in our partial results on mutations of simple-minded objects. These results can also potentially serve as new tools to the study of the long-standing Auslander-Reiten Conjecture.
In another direction, the work carried out with Karin Baur on the geometric model for string algebras, which is currently in preparation for publication, widely extends the current state of the art, where a lot of research was carried out for gentle algebras establishing connections with other areas in mathematics, such as symplectic geometry.
The project on maximal almost rigid objects, which started during my participation in a research programme in Cambridge in September 2021, is current work in progress and is likely to lead to two publications and to provide a new connection between gentle algebras and Cambrian combinatorics.
In another direction, the work carried out with Karin Baur on the geometric model for string algebras, which is currently in preparation for publication, widely extends the current state of the art, where a lot of research was carried out for gentle algebras establishing connections with other areas in mathematics, such as symplectic geometry.
The project on maximal almost rigid objects, which started during my participation in a research programme in Cambridge in September 2021, is current work in progress and is likely to lead to two publications and to provide a new connection between gentle algebras and Cambrian combinatorics.