We initiate the study of Boolean function analysis on high-dimensional expanders, and more generally, on weighted simplicial complexes.
We identify a linear-algebraic condition under which there is a notion of Fourier expansion for functions on the facets of the complex, a notion similar to the unique harmonic multilinear expansion for functions on the slice or Johnson scheme, and sharing many of its properties.
We prove a rudimentary FKN theorem for high-dimensional expanders, utilizing a recent agreement theorem of Dinur and Kaufman.
Our work also gives a novel definition of high-dimensional expansion. We also generalize this notion (work in progress) to the Grassmann scheme.