When its really just a Matrix!

Pierce decompositions provide a rich collection of tools to the decomposition tool box. Often shrouded by notation and big words they answer a rather simple and well motivated question: "Is this thing really just a matrix if I squint hard enough?" and often the answer is yes!

Lets look at an example of what we are after: Matrices. Let $M_2(\mathbb{R})$ be the $2$-by-$2$ matrices over $\mathbb{R}$ and notice that we can decompose any matrix into into components, meaning $$M_2(\mathbb{R})=\left\{\begin{bmatrix}a & b\\ c & d\end{bmatrix} \Big| a,b,c,d\in\mathbb{R} \right\}=\left\{\begin{bmatrix}a & 0\\ 0 & 0\end{bmatrix} \right\}\oplus\left\{\begin{bmatrix}0 & b\\ 0 & 0\end{bmatrix} \right\}\oplus \left\{\begin{bmatrix}0 & 0\\ c & 0\end{bmatrix} \right\}\oplus \left\{\begin{bmatrix}0 & 0\\ 0 & d\end{bmatrix} \right\}$$ So our motivating question is this: When is an arbitrary Algebra decomposable into what looks like a behaves like the components of a matrix? And when is multiplication really just matrix multiplication?

The History

ill skip this for now but ill comment that the Krull-Schmidt theorem was first proven by Wedderburn, who doesnt not get their names on the theorem.

Why Study Representations?

connect to modules. idk give top of triangle of representation (what James calls Cayley's Resonance)

Why Decompose Representations?

What is a decompose?

How to Decompose: A Recursive Algorithm to Decompose Representations

Lets first look at an example of a decomposition to see what we should being looking for. Consider the $2$-dimensional vector space $V=K^2$ over your favorite field $K$. This can be thought of as an action by scalar multiplication on $V$, where $K\times V\rightarrow V$. Its easy to see that $V$ decomposed into two subspaces on its components: $$K^2=\left\{\begin{bmatrix}*\\0\end{bmatrix}\right\}\oplus \left\{\begin{bmatrix}0\\ *\end{bmatrix}\right\}\cong K\oplus K$$

This is a rather boring example but it can still help highlight what we are after, in the sense that each of these factors can be thought of as coming from a projection. This idea can be highlighted in following diagram:$$j$$

This is telling us that in order to find decompositions we need to find idempotents. idk what to put here but we get a correspondence between decompositions of modules and a collection of idempotent maps on $V$

Theorem: There is a 1-to-1 correspondence between collections of idempotents $\mathcal{E}\subseteq\text{End}(V)$ and direct decompositions of $V$