Let $I$ be a finite poset.

Definition: A representation of $I$ is a functor $I\to\mathrm{Vect}_{\mathbb C}\ $. Equivalently, a representation of $I$ is a module over its incidence algebra $\mathbb C I$.

The posets of finite representation type have been determined by Michele Loupias in the very brief paper Indecomposable representations of finite ordered sets.

Request: Is there a more comprehensive discussion of finite representation type posets (ideally including explicit descriptions of indecomposable representations) somewhere in the literature?

There is a related but different notion called matrix representation in Daniel Simson: Linear representations of partially ordered sets and vector space categories:

Definition: A matrix representation of $I=\{1,\ldots,n\}$ (equipped with a partial order $\preceq$) consists of vector spaces $V_1,\ldots,V_{n+1}$ and linear maps $t_j\colon V_j\to V_{n+1}$ for $J=1,\ldots,n$. A morphism of matrix representations is a commutative diagram $$ \begin{array}{ccc} V_1\oplus\cdots\oplus V_n & \xrightarrow{(t_j)} & V_{n+1} \\ \downarrow & & \downarrow\\ V_1^'\oplus\cdots\oplus V_n^' & \xrightarrow{(t_j^')} & V_{n+1}^' \end{array} $$ in which the map $V_i\to V_j^'$ vanishes if $i\not\preceq j$.

The category of matrix representations of $I$ turns out to be equivalent to a full subcategory of the category of $\mathbb C I^*$-modules, where $I^*$ is the poset obtained from $I$ be adding a unique maximal element.

Question: Given that the notion of representation is more natural than the notion of matrix representation (at least to me), why are matrix representations so much more prominent in the literature?


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