Representation theory and symmetry

A group G consists of elements with an associative composition, an identity, and inverses. A representation over a field sends each element to an invertible linear map on a vector space V, preserving multiplication and identity. If applying h and then g is written gh, the corresponding matrix product must use the same order. Coordinates only describe the linear maps; changing a basis conjugates every matrix by the same invertible matrix. Two such descriptions are equivalent representations, even if individual entries look different.

The kernel records group elements acting as the identity on every vector. A faithful representation has trivial kernel, so it distinguishes group elements. That property says nothing about whether the vector space splits into simpler invariant pieces. For the group with two elements, a reflection matrix acting on the plane is faithful because the nonidentity element changes some vectors. Nevertheless its fixed axis and reversed axis are separate one-dimensional invariant subspaces. Recovering the symmetry and decomposing its action are distinct questions with distinct tests.

A subspace W is invariant when every represented group element sends W into itself. The zero subspace and the whole space always qualify. A nonzero representation is irreducible when there are no other invariant subspaces. This is stronger than asking whether one chosen matrix has distinct eigenvalues, because the same subspace must work for the entire group. For commuting complex matrices, common invariant directions often help; for noncommuting actions, an eigenline of one element may be moved away by another element.

A decomposition into invariant summands lets each operation be written in blocks. For finite groups over the complex numbers, averaging an inner product over all group elements makes it invariant. The orthogonal complement of an invariant subspace is then invariant too, yielding complete reducibility in finite dimension. This argument divides by the group order and relies on a positive definite complex inner product. In a field whose characteristic divides that order, analogous splitting can fail. Likewise representations of infinite groups do not inherit this finite averaging proof merely because their matrices are small.

The character assigns to a group element the trace of its action. Trace is unchanged by simultaneous basis conjugation and is constant on conjugacy classes. It adds under direct sums and multiplies under tensor products. For finite-dimensional complex representations of finite groups, character theory determines multiplicities of irreducible summands. This powerful classification statement is not a universal assertion about every field, every infinite group, or arbitrary nonsemisimple algebra representations. Always keep the setting attached to the theorem rather than treating a useful numerical summary as complete by definition.

To study a new symmetry action, first write the group relations and check them on the proposed matrices. Then locate invariant subspaces and test whether a direct complement exists under the available hypotheses. Finally use traces to compare decompositions without performing a fresh basis change for every element. The vector lab below supplies only one rotating vector and its projections; it helps distinguish a vector from its coordinate description. It does not enumerate a group, identify irreducibles, or certify that a whole family of matrices respects group multiplication.