subrepresentation

In representation theory, a subrepresentation of a representation V of a group G (or of an algebra) is a subspace W of V that is stable under the action of G. Equivalently, for all g in G and w in W, g·w is again in W. When this holds, W with the restricted action of G is itself a representation of G, called a subrepresentation of V.

If W is a subrepresentation of V, the quotient space V/W inherits a G-action defined by g·(v + W) = gv + W. This makes V/W a G-representation as well. Subrepresentations correspond to invariant subspaces, and the inclusion map W ↪ V is a G-linear (intertwining) map.

A representation V is called irreducible (or simple) if it has no nontrivial proper subrepresentations. The study of subrepresentations leads to a decomposition theory: under suitable conditions, such as Maschke’s theorem for finite groups over fields of characteristic zero or not dividing the group order, every representation decomposes as a direct sum of irreducible subrepresentations and, dually, every invariant subspace is a direct summand.

The set of subrepresentations of V forms a lattice under inclusion, reflecting the internal structure of the representation. Subrepresentations are central to constructing and classifying representations, via concepts like composition series and Jordan–Hölder constituents. In category terms, subrepresentations are subobjects in the category of G-modules, and simple subrepresentations correspond to the building blocks of more complex representations.

Examples: in the permutation representation of a group on a vector space, invariant lines or planes are subrepresentations. For instance, the line spanned by a fixed vector invariant under all group actions, or the standard representation of the symmetric group obtained by the orthogonal complement of the trivial subrepresentation.