Selfconjugate

Self-conjugate refers to an object that is invariant under a conjugation operation, meaning the object is equal to or isomorphic to its conjugate. The term appears in several areas of mathematics, each with its own precise sense.

In complex numbers, a number z is self-conjugate if z equals its complex conjugate z̄. This occurs precisely when z is real, since z̄ = z̄ complex conjugation changes the sign of the imaginary part.

In linear algebra and representation theory, a representation or matrix may be called self-conjugate if it is isomorphic to its complex conjugate representation or matrix. For a group representation, V is self-conjugate if V is isomorphic to the complex conjugate representation V̄; equivalently, its character χ takes only real values. The notion connects to the broader idea of self-duality, as a self-conjugate representation often admits a real or quaternionic form, distinguished by the Frobenius–Schur indicator (positive for a real form, negative for a quaternionic form, zero when the representation is not self-conjugate).

In combinatorics, a self-conjugate partition is a partition whose Ferrers diagram is symmetric with respect to the main diagonal, i.e., it is equal to its conjugate partition. Such partitions have diagonal length equal to the number of parts and are counted by partitions into distinct odd parts; their generating function is the infinite product ∏_{k≥1} (1+q^k).

Across contexts, self-conjugate thus describes invariance under a conjugation operation, whether applied to numbers, representations, partitions, or matrices.