A new basis for generalized harmonic polynomials, with application to quantum mechanics

What is it about?

So-called SN-harmonic polynomials are a generalization of standard harmonic polynomials, such that they are annihilated by symmetrized derivatives of any order, not just the (second-order) Laplacian. They are most easily described as all possible derivatives of the Vandermonde determinant. The vector spaces spanned by these polynomials are of classical interest in combinatorics and representation theory. Here a new basis is found for these spaces with good symmetry properties, in particular, each basis vector is indexed by a standard ribbon Young tableau.

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Why is it important?

The basis found in this work is a stepping stone to constructing generating sets of N-fermion wave functions, called shapes. Shapes are a finite number of antisymmetric wave functions such that all antisymmetric wave functions can be constructed from them, with symmetric-function (bosonic) coefficients. They open a unique perspective on the geometry of many-body Hilbert space, such that strong correlations in real space are geometric (kinematic) constraints in Hilbert space. In particular, bands in the spectra of finite systems are interpreted algebraically as ideals generated by individual shapes.

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