To the bicommutant theorem for algebras generated by symmetries of finite point sets in $\mathbb{R}^3$

The problem of describing invariant extensions of the 3D Schrödinger operator $\mathbf{H}$ with a finite number of point interactions leads to the need for studying matrices of a special type, the permutation matrices. A large class of such extensions considered in a certain boundary triplet is in one-to-one correspondence with a set of the so-called boundary operators (matrices). The extension of the operator $\mathbf{H}$ with point interactions concentrated on $X = \{x_1, \ldots, x_m\}$ is invariant under the symmetry group of $X$ (or its subgroup) if and only if the corresponding boundary matrix commutes with the set of permutation matrices of size $m\times m$ induced by the symmetry group, i.e., belongs to the commutant of this set. The bicommutant theorem for such a set of matrices is proved for an arbitrary finite point set. For some special cases – a regular polygon, a tetrahedron, and a cube – the basis for the bicommutant regarded as a vector space is given explicitly.