Modified diagonals and linear relations between small diagonals

Abstract: We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the $\mathbb{Z}$-linear relations between small $m$-diagonals $\text{pt}^{{1,\ldots,n}\setminus A}\times\Delta_A$ in the rational Chow ring of $X^n$ for $A$ ranging over $m$-element subsets of ${1,\ldots,n}$. Our results generalize to arbitrary symmetric classes in place of the diagonal in $X^m$, and with different types of inclusions $A^\bullet(X^m)_{\mathbb{Q}}^{S_m} \hookrightarrow A^\bullet(X^n)_{\mathbb{Q}}$. The combinatorial heart of this paper, which may be of independent interest, is showing the $\mathbb{Z}$-linear relations between elementary symmetric polynomials $e_k(x_{a_1},\ldots,x_{a_m}) \in \mathbb{Z}[x_1,\ldots,x_n]$ are generated by the $S_n$-translates of a certain alternating sum over the facets of a hyperoctahedron.