  The Sachdev-Ye-Kitaev (SYK) model, in its simplest form, describes $k$
Majorana fermions with random all-to-all four-body interactions. We consider
the SYK model in the framework of a many-body Altland-Zirnbauer classification
that sees the system as belonging to one of eight (real) symmetry classes
depending on the value of $k\mod 8$. We show that, depending on the symmetry
class, the system may support exact many-body zero modes with the symmetries
also dictating whether these may have a nonzero contribution to Majorana
fermions, i.e., single-particle weight. These zero modes appear in all but two
of the symmetry classes. When present, they leave clear signatures in physical
observables that go beyond the threefold (Wigner-Dyson) possibilities for level
spacing statistics studied earlier. Signatures we discover include a
zero-energy peak or hole in the single-particle spectral function, depending on
whether symmetries allow or forbid zero modes to have single-particle weight.
The zero modes are also shown to influence the many-body dynamics, where
signatures include a nonzero long-time limit for the out-of-time-order
correlation function. Furthermore, we show that the extension of the four-body
SYK model by quadratic terms can be interpreted as realizing the remaining two
complex symmetry classes; we thus demonstrate how the entire tenfold
Altland-Zirnbauer classification may emerge in the SYK model.
