$\tau$-tilting theory & the physics of scattering amplitudes - Hugh Thomas
Ellis Caird
Created: 20-11-24
1.1) Generic Fan decomposition
Looking at 'generic behaviour' of quiver reps of a fixed dimension vector which roughly means behaviour that holds off of solutions to polynomial equations i.e. on a Zariski open set inside of $\mathrm{Rep}_Q(\underline{d})$. Exact definition of 'generic' is still a little elusive; in theory there are many such zariski open subsets(?) which is the 'right' one? Look at $A_2$ first
$A_2: 1 \overset{\alpha}{\longleftarrow} 2$
For $\underline{d} = (1,0)$ we have no real choice but $\mathbb{C} \overset{0}{\longleftarrow} 0$.
For $\underline{d} = (1,1)$ we have two choices up to isomorphism,
$\mathbb{C} \overset{\alpha}{\longleftarrow} \mathbb{C}$ for $\alpha \neq 0$ and $\mathbb{C} \overset{0}{\longleftarrow} \mathbb{C}$.
The latter is $S_1 \oplus S_2$ and we'll denote the isomorphism class (which is Zariski open) of the former by $M_{1,1}$.
For $\underline{d} = (1,1)$ we have a choice of a $d \times d$ complex matrix and the generic condition here is determinant being non-zero in which case we get $M_{1,1}^d$.
For $\underline{d} = (d_1, d_2)$ with $d_1 > d_2$ we are choosing a $d_1 \times d_2$ matrix and the generic choice here is an injection which subequently will split the rep as $M_{1,1}^{d_2} \oplus S_1^{d_1 - d_2}$. For the other order we will genericall have a surjective that will split as a direct sum of copies of $S_2$ and $M_{1,1}$. This yields a nice summarising picture:
1.2) The Dynkin case
- Dynkin quivers
- Roots systems
- Gabriel's theorem
- $\operatorname{dimExt}^1(X,X)$ = codim of isoclass of X in Rep_Q(d)
Corollaries:
- Indecomposable's from Gabriel's theorem are rigid
- Unique way to decompose a dim vec into a sum of dim vecs whose generic reps have no extensions between them.
A cone in a real vector space is all non-negative linear combinations of a set of fixed vectors. Relative interior: points in cone but not on any proper face.
A cone is simplicial if the generators of its rays form a basis.
A fan is a collection of cones thats that is closed under taking faces and intersections.
Given a quiver, Q, we define a fan $\Sigma_Q$ where the rays are generated by the positive roots of $Q$ and the cones are generated by subsets of positive roots corresponding to reps with no extensions between eachother. This cone is simplicial.