Representation of a partially ordered set

Let be a partially ordered set and a field. Let be a symbol with . An -space is of the form , where the are subspaces of the -space for , such that implies . Let be -spaces; a mapping is a -linear mapping such that for all . The direct sum of and is with for all . An -space is said to be indecomposable if it cannot be written as the direct sum of two non-zero -spaces.

The partially ordered set is called subspace-finite if there are only finitely many isomorphism classes of indecomposable -spaces. Kleiner's theorem asserts that is subspace-finite if 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