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The AR quiver of the regular representations of an affine quiver consists of infinitely many "tubes". A tube of rank $r$ has $r$ modules on what you call the border. Let me number them $B_1, B_2, \dot …

There is a beautiful interpretation of $f(\chi)$ (that is to say, of the length of the first column of the partition), though it isn't very representation-theoretic. One way to generate Plancherel mea …

For $\mathcal A$-cluster algebras of finite type, it is very easy to describe the theta-basis: it consists of the cluster monomials. Is there any similarly easy way to describe the theta-basis for $\m …

The quiver given in the question has five simple modules, six which correspond to a single arrow, and the remaining representations have support as follows: 123, 124, 125, 235, 345, 1235, 12235 (note …

One example of an elementary application of cluster algebras is the proof that the Somos-4 and Somos-5 sequences, which are defined by a simple recursion, are integral. This is so because the entries …

For the Kronecker quiver (two vertices, two arrows in the same direction) and dimension vector (1,1), over an algebraically closed ground field, the indecomposables are naturally parameterized by poin …

Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at th …

I think there is good reason to think the answer is "no". In rank 2, the theta basis agrees with the greedy basis (arXiv:1508.01404). Greedy basis elements are indecomposable positive elements (see …

Let $Q$ be a quiver, and let $d=(d_i)$ be a dimension vector. We can consider Rep($Q,d$), the affine space consisting of representations of $Q$ with dimension vector $d$. The general linear $GL(d)= \p …

It turns out the answer is "no". There is an example in section 3.4 of Riedtmann's paper "Degenerations for representations of quivers with relations", Ann. sci. Éc. Norm. Sup. v. 19 (1986), 275-301. …

This answer perhaps says things that are all obvious to the OP. $\textrm{Ext}^i(A,B)$ is a vector space over the ground field, so its cardinality is $q^d$ where $d$ is the dimension of the vector sp …

There is relevant information here, including a statement of the conjecture (as Conjecture 5.1). http://www.math.uni-bonn.de/people/schroer/fd-problems-files/FD-RigidModulesConj.pdf That preprint …

Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the positiv …

Let $\mathcal O$ be the structure sheaf of $\mathbb P^1$. Then $\mathcal O \oplus \mathcal O(1)$ is rigid and generates the derived category of coherent sheaves on $\mathbb P^1$. Thus, it is a tilti …

I am going to give a negative answer for the first question, under a stronger notion of canonicity. The approach I want to take is to consider the poset of tilting modules. They are ordered by in …