  We generalize the Jacobi no-core shell model (J-NCSM) to study
double-strangeness hypernuclei. All particle conversions in the strangeness
$S=-1,-2$ sectors are explicitly taken into account. In two-body space, such
transitions may lead to the coupling between states of identical particles and
of non-identical ones. Therefore, a careful consideration is required when
determining the combinatorial factors that connect the many-body potential
matrix elements and the free-space two-body potentials. Using second
quantization, we systematically derive the combinatorial factors in question
for $S=0,-1,-2$ sectors. As a first application, we use the J-NCSM to
investigate $\Lambda \Lambda$ s-shell hypernuclei based on hyperon-hyperon (YY)
potentials derived within chiral effective field theory at leading order (LO)
and up to next-to-leading order (NLO). We find that the LO potential overbinds
$^{\text{ }\text{ }\text{ } \text{}6}_{\Lambda \Lambda}\text{He}$ while the
prediction of the NLO interaction is close to experiment. Both interactions
also yield a bound state for $^{\text{ }\text{ }\text{ } \text{}5}_{\Lambda
\Lambda}\text{He}$. The $^{\text{}\text{ }\text{ }\text{}4}_{\Lambda
\Lambda}\text{H}$ system is predicted to be unbound.
