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Odd-frequency superconductivity describes a class of superconducting stateswhere the superconducting gap is an odd function in relative time and Matsubarafrequency. We present a group theoretical analysis based on the linearized gapequation in terms of Shubnikov groups of the second kind. By discussing systemswith spin-orbit coupling and an interaction kernel which is symmetric under thereversal of relative time, we show that both even- and odd-frequency gaps areallowed to occur. Specific examples are discussed for the square lattice, theoctahedral lattice, and the tetragonal lattice. For irreducible representationsthat are even under reversal of relative time the common combinations of $s$-and d-wave spin singlet and p-wave spin triplet gaps are revealed, irreduciblerepresentations that are odd under reversal of relative time give rise to $s$-and d-wave spin triplet and p-wave spin singlet gaps. Furthermore, we discussthe construction of a generalized Ginzburg-Landau theory in terms of theassociated irreducible representations. The result complements the establishedclassification of superconducting states of matter.

Symmetry analysis of odd- and even-frequency superconducting gap symmetries for time-reversal symmetric interactions