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