Representation of a partially ordered set

Let $S$ be a partially ordered set and $k$ a field. Let $\omega$ be a symbol with $\omega \notin S$. An $S$- space is of the form $V=( V _ \omega , V _ {s} ) _ {s \in S }$, where the $V _ {s}$ are subspaces of the $k$- space $V _ \omega$ for $s \in S$, such that $s \leq s ^ \prime$ implies $V _ {s} \subset V _ {s ^ \prime }$. Let $V , V ^ \prime$ be $S$- spaces; a mapping $f: V \rightarrow V ^ \prime$ is a $k$- linear mapping $V _ \omega \rightarrow V _ \omega ^ \prime$ such that $f( V _ {s} ) \subset V _ {s} ^ \prime$ for all $s \in S$. The direct sum of $V$ and $V ^ \prime$ is $V \oplus V ^ \prime$ with $( V \oplus V ^ \prime ) _ {s} = V _ {s} \oplus V _ {s} ^ \prime$ for all $s \in S \cup \{ \omega \}$. An $S$- space is said to be indecomposable if it cannot be written as the direct sum of two non-zero $S$- spaces.

The partially ordered set $S$ is called subspace-finite if there are only finitely many isomorphism classes of indecomposable $S$- spaces. Kleiner's theorem asserts that $S$ is subspace-finite if $S$ is finite and does not contain as a full subset one of the partially ordered sets

Figure: r081400a

see [a1]. M.M. Kleiner also has determined all the indecomposable representations of a representation-finite partially ordered set [a2]. A characterization of the tame partially ordered sets has been obtained by L.A. Nazarova [a3]. The representation theory of partially ordered sets plays a prominent role in the representation theory of finite-dimensional algebras.

References

[a1]	M.M. Kleiner, "Partially ordered sets of finite type" J. Soviet Math. , 3 (1975) pp. 607–615 Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. , 28 (1972) pp. 32–41
[a2]	M.M. Kleiner, "On the exact representations of partially ordered sets of finite type" J. Soviet Math. , 3 (1975) pp. 616–628 Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. , 28 (1972) pp. 42–60
[a3]	L.A. Nazarova, "Partially ordered sets of infinite type" Math. USSR Izv. , 9 : 5 (1975) pp. 911–938 Izv. Akad. Nauk SSSR Ser. Mat. , 39 (1975) pp. 963–991