  The basic question is addressed, how the space dimension $d$ is encoded in
the Hilbert space of $N$ identical fermions. There appears a finite number
$N!^{d-1}$ of many-body wave functions, called shapes, which cannot be
generated by trivial combinatorial extension of the one-dimensional ones. A
general algorithm is given to list them all in terms of standard Slater
determinants. Conversely, excitations which can be induced from the
one-dimensional case are bosonised into a system of distinguishable bosons,
called Euler bosons, much like the electromagnetic field is quantized in terms
of photons distinguishable by their wave numbers. Their wave functions are
given explicitly in terms of elementary symmetric functions, reflecting the
fact that the fermion sign problem is trivial in one dimension. The shapes act
as vacua for the Euler bosons. They are the natural generalization of the
single-Slater-determinant form for the ground state to more than one dimension.
In terms of algebraic invariant theory, the shapes are antisymmetric invariants
which finitely generate the $N$-fermion Hilbert space as a graded algebra over
the ring of symmetric polynomials. Analogous results hold for identical bosons.
