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

Natural generalization of the ground-state Slater determinant to more than one dimension