Applications of representation theory in topology

Question

I'm beginning to study representation theory, in particular, quiver representations. Since I have more familiarity with topology, I was wondering if there is any applications of these things to algebraic topology. So far, I was able to find none.

Answer 1

Here is an application of quiver representation theory to the study of persistent homology.

First, a quick recap on persistent homology: starting with some data (e.g., a point cloud), we can associate a filtered topological space/simplicial complex $$X_1 \to X_2 \to \cdots \to X_n.$$ Applying homology with coefficients in a field $\mathbb{F}$ to such an object yields a filtered vector space $\{V_i := H_k(X_i)\}_{i=1}^n$ which we can call a persistence module: $$V_1 \to V_2 \to \cdots \to V_n.$$ In applications, it is convenient to obtain a compact summary of these persistence modules. Regarding such modules as modules over the PID $\mathbb{F}[t]$, we have a structure theorem that tells us these modules can be decomposed into a direct sum of so-called interval modules, which are completely described by when a certain homology class appears and disappears in the filtration.

Later on, people became interested in persistence modules that are indexed by other diagrams. For example, there is a notion of zigzag persistence, where we start with a diagram of spaces like before, but in which some arrows go "backwards", e.g.: $$V_1 \to V_2 \leftarrow V_3 \to \cdots \leftarrow V_n.$$ Another example are the so-called multidimensional persistence modules, in which the indexing diagram is "higher dimensional" (and we require the squares to commute as well): $$\require{AMScd} \begin{CD} V_{11} & \rightarrow & V_{12} & \rightarrow & \cdots & \rightarrow & V_{1n} \\ \downarrow & & \downarrow &&&& \downarrow \\ V_{21} & \rightarrow & V_{22} & \rightarrow & \cdots & \rightarrow & V_{2n} \\ \downarrow & & \downarrow & & & & \downarrow \\ \vdots & & \vdots & & \ddots & & \vdots \\ \downarrow & & \downarrow & & & & \downarrow \\ V_{m1} & \rightarrow & V_{m2} & \rightarrow & \cdots & \rightarrow & V_{mn} \end{CD}$$

The question now is: do such "generalized" persistence modules admit simple descriptions? If you've seen quiver representations before, you can probably recognize the ordinary and zigzag-type persistence modules as nothing more than just representations of the $A_n$ quiver. The tractability of such quiver representations (e.g., there is a finite number of indecomposables) follows from the fact that $A_n$ is of finite type.

Can all generalized persistence modules (say over a finite diagram) be decomposed into a sum of indecomposables, of which there are only finitely many? If so, this would be great because we could describe any generalized persistence module simply by giving a finite list consisting of the multiplicities of each of the finitely many indecomposables. Alas, Gabriel's theorem tells us that finite type quivers are quite special: only those quivers whose underlying graph is a Dynkin diagram are of finite type.

It turns out that the multidimensional persistence modules are also not of finite type. OK, so maybe asking for finitely many indecomposables is too much. We might be satisfied if the indecomposables can be organized into one-parameter families; such quivers are called tame. This leads us into the trichotomy of finite-type, tame, and wild quivers. Unfortunately, the theory of 2D persistence modules is wild in general, so these representations can be extremely complicated. There is a lot of ongoing research into understanding them better and incorporating them into applications.

Edited to add: a good reference for the interplay between quiver representations and persistence theory is Steve Oudot's book Persistence Theory: From Quiver Representations to Data Analysis, available on his website.