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

Modified diagonals and linear relations between small diagonals