Aspects of representation theory: τ-exceptional sequences, modular Fuss-Catalan numbers and idempotent completion of extriangulated categories

Abstract This thesis is concerned with various aspects of the representation the- ory of finite dimensional algebras, with a focus on combinatorial and homological aspects. We explore the aspects of representation theory relating to tilting modules, cluster algebras, τ-exceptional sequences, and extriangulated categories. The notion of a τ-exceptional sequence was introduced by Buan and Marsh in Buan & Marsh (2021) as a generalisation of an exceptional sequence for finite-dimensional algebras. We calculate the number of complete τ-exceptional sequences of certain classes of Nakayama al- gebras. In some cases, we obtain closed formulas which also count other well-known combinatorial sets and exceptional sequences of path algebras of Dynkin quivers. The modular Catalan numbers C(k,n), introduced in Hein & Huang (2017) count equivalence classes of parenthesizations of x0 ∗ · · · ∗ xn, where ∗ is a binary k-associative operation and k is a positive inte- ger. The classical notion of associativity coincides with 1-associativity, in which case C(1,n) = 1, and the single 1-equivalence class has size given by the Catalan number Cn. We introduce modular Fuss-Catalan numbers Cm which count k-equivalence classes of parenthesizations of k,n x0 ∗ · · · ∗ xn where ∗ is an m-ary k-associative operation for m ≥ 2. Our main results are an explicit formula for Cm , and a characterisation of k-associativity. Extriangulated categories were introduced by Nakaoka and Palu in Nakaoka & Palu (2019a) as a simultaneous generalisation of exact cat- egories and triangulated categories. We show that the idempotent completion of an extriangulated category is also extriangulated. A possible consequence of this is a methodology for constructing Krull- Remak-Schmidt extriangulated categories, since an additive category A has the Krull-Remak-Schmidt property if and only if A is idempotent complete and the endomorphism ring of every object is semi-perfect; see (Krause, 2015, Corollary 4.4).