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Murali's basis for the slice

Yuval Filmus and Nathan Lindzey

The slice consists of all vectors $(x_1,\ldots,x_n) \in \{0,1\}^n$ of Hamming weight $k$. We usually assume that $k \leq n/2$.

My paper constructs an explicit orthogonal basis for functions on the slice. The basis is canonical in a sense made clear by Murali K. Srinivasan, who had given an inductive construction for the same basis in his own paper.

In this short note, we first show how Murali’s inductive construction gives rise to the explicit formula appearing in my paper. Then, we briefly give yet another construction of the basis, using the Gram–Schmidt process.