A torsion class is a collection of modules for an algebra that is closed under quotients and extensions. It turns out that many combinatorial objects can be realized as torsion classes for certain algebras, and the containment order on torsion classes recovers interesting partial orders on those objects. We will give a brief introduction to several examples, including the Tamari lattice, the weak order, and the oriented exchange graph of a cluster algebra. We will also outline the schedule for the first half of the semester.

We will define quivers and their representations and use examples to gain intuition. We will then define algebraic objects associated to quivers (the path algebra and the \(K\)-group) and use these to state the standard representation of a quiver and explore properties of the category of quiver representations. This talk will cover sections 1.1–1.4 of Quiver Representations and Quiver Varieties by Alexander Kirillov Jr.

We review notions from last lecture, including projective and simple modules over the path algebra \(kQ\) of a quiver. From the projective resolution of any module over \(kQ\), we compute dimensions of Hom and Ext functors. We use this to motivate the definition of the Euler form, classify when the symmetrized Euler form is positive (semi)definite, and draw connections to Lie theory. We finally state the definition and classification of finite type quivers. This talk will cover sections 1.4–1.7 and 3.1 of Quiver Representations and Quiver Varieties by Alexander Kirillov Jr.

The protagonist of this talk will be the oriented quiver of Dynkin type A. We will introduce its (finitely many) indecomposable representations, use them to classify all quotient-closed subcategories of rep \(A_n\), and discover which among these constitute torsion classes via “bracket vectors”. The torsion classes ordered by inclusion recover the Tamari lattice, a beloved combinatorial lattice encoding a variety of known objects. This talk follows the paper “The Tamari lattice as it arises in quiver representations” by Hugh Thomas.

Last time we saw that we can describe torsion classes of quiver representations of Type \(A_n\) by bracket vectors. In this talk, I will give some more general approaches to describing torsion classes, First I will illustrate them through the familiar Type \(A_n\) example. I will also demonstrate how to use them to describe the torsion classes of the representations of the Kronecker quiver. In the second part of the talk, I will use these new descriptions to construct a dual notion of torsion classes, the torsion free classes, and prove that torsion classes form a complete lattice.

Last week, Yucong introduced the lattice of torsion classes. This coming week, we’ll examine several properties of the lattice of torsion classes. We’ll characterize the completely join irreducible elements and, time permitting, prove that the lattice of torsion classes is completely semi-distributive. Underlying both of these properties are “bricks”, a special kind of indecomposable module.