  Many-body Hilbert space is a functional vector space with the natural
structure of an algebra, in which vector multiplication is ordinary
multiplication of wave functions. This algebra is finite-dimensional, with
exactly $N!^{d-1}$ generators for $N$ identical particles, bosons or fermions,
in $d$ dimensions. The generators are called shapes. Each shape is a possible
many-body vacuum. Shapes are natural generalizations of the ground-state Slater
determinant to more than one dimension. Physical states, including the ground
state, are superpositions of shapes with symmetric-function coefficients, for
both bosons and fermions. These symmetric functions may be interpreted as
bosonic excitations of the shapes. The algebraic structure of Hilbert space
described here provides qualitative insights into long-standing issues of
many-body physics, including the fermion sign problem and the microscopic
origin of bands in the spectra of finite systems.
