A Fast Fourier Transform for the Johnson Graph

Abstract

The set X of k-subsets of an n-set has a natural graph structure where two k-subsets are neighbors if and only if the size of their intersection is \(k-1\). This is known as the Johnson graph. The symmetric group \(S_n\) acts on the space of complex functions on X and this space has a multiplicity-free decomposition as sum of irreducible representations of \(S_n\), so it has a well-defined Gelfand–Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand–Tsetlin basis. The direct application of this matrix to a generic vector requires \(\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\) arithmetic operations. We show that, in analogy with the classical Fast Fourier Transform on the discrete circle, this matrix can be factorized as a product of \(n-1\) orthogonal matrices, each one with at most two nonzero elements in each column. The factorization is based on the construction of \(n-1\) intermediate bases which are parametrized via the Robinson–Schensted insertion algorithm. This factorization shows that the number of arithmetic operations required to apply this matrix to a generic vector is bounded above by \(2(n-1) \left( {\begin{array}{c}n\\ k\end{array}}\right) \). We give an algorithm that constructs all these factors using at most \(289(n-1)\left( {\begin{array}{c}n\\ k\end{array}}\right) \) arithmetic operations. The coefficients of these matrices are rational numbers and the construction does not depend on numerical methods. Instead, they are obtained by solving small linear systems with integer coefficients derived from the Jucys–Murphy operators. In particular we avoid the use of square roots. As a consequence, we show that the problem of computing all the weights of the isotypic components of a given function can be solved in \(O(n \left( {\begin{array}{c}n\\ k\end{array}}\right) )\) operations, improving the previous bound \(O(k^2 \left( {\begin{array}{c}n\\ k\end{array}}\right) )\) when k asymptotically dominates \(\sqrt{n}\). The same improvement is achieved for the problem of computing the isotypic projection onto a single component.