Aaron Chan

We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra $T(A)$ of a $d$-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that $d$-representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.

We initiate the investigation of representation theory of non-orientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked non-orientable surfaces, we study a certain modification on the objects of the cluster category associated to the orientable double covers in the unpunctured case. More precisely, we consider symmetric representation theory studied by Derksen-Weyman and Boos-Cerulli Irelli, and lift it to the cluster category. This gives a way to consider `indecomposable orbits of objects' under a contravariant duality functor. Hence, we can assign curves on a non-orientable surface $(\mathbb{S}, \mathbb{M})$ to indecomposable symmetric objects. Moreover, we define a new notion of symmetric extension, and show that the arcs and quasi-arcs on $(\mathbb{S}, \mathbb{M})$ correspond to the indecomposable symmetric objects without symmetric self-extension. Consequently, we show that quasi-triangulations of $(\mathbb{S}, \mathbb{M})$ correspond to a symmetric analogue of cluster tilting objects.

In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called $p$-standard tableaux.