M/E23

Broadly speaking, representation theory focuses on classifying modules over associative algebras up to isomorphism. The representation type of an algebra reflects the complexity of this classification problem. For algebras over an algebraically closed field, a famous theorem by Y. Drozd divides finite-dimensional algebras into three representation types: finite, tame, and wild. A more rough notion of g-representation type is currently being envisioned. The aim is then to classify the so-called $\tau$-rigid modules, which have very nice combinatorial properties. These modules are key in defining the g-vector fan, a polyhedral fan in Euclidean space. The g-representation type of an algebra is linked to the properties of this fan. In this talk, we investigate when traditional representation types and g-representation types coincide. Our main result focuses on incidence algebras of posets. We outline a proof that, for any finite poset, its incidence algebra is of finite representation type if and only if it is of g-finite representation type. If the poset is also simply connected, the incidence algebra is of tame representation type if and only if it is of g-tame representation type. Our original results are joint with J. F. Grevstad and E. S. Rundsveen [arXiv:2407.17965].

Erlend Børve

Köln