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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 …

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 …

One simple answer is to talk about the totally positive part of $(G_{k,n})_{> 0}$, the part of the Grassmannian where all the maximal minors (=Plücker coordinates) are real and positive. Naively, if …

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 …

Jan's answer includes many excellent references. I will try to give a few quick comments. First of all, although the original Buan-Marsh-Reineke-Reiten-Todorov paper contained some results which wer …

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 …

Neither of the references linked in the comments seem to solve the OP's question. Gabriel's theorem says that the indecomposables correspond to positive roots. The way this correspondence works is a …

This sounds wrong to me. I think $D^b(Q)$ should be replaced by the derived category of finite length modules over the corresponding preprojective algebra of affine type. Homological mirror symmetry …

I once mentioned in a talk (to a group of algebraic combinatorialists) that "A quiver is just a directed graph". An audience-member stuck up his hand to say "A quiver is a directed graph with pretens …

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 …

There is another way to relate representations of $Q$ to representations of $Q'$: reflection functors. These are quite easy to describe combinatorially. One downside is that the way they work is by …

First of all, you have to assume that $A$ is non-semi-simple. For a semi-simple Artin algebra, the representation dimension is defined to be 1. For a non-semi-simple algebra, the representation di …

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 …

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 …