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 ${\text{pt}}^{\{1,\dots ,n\}\setminus A}\times {\Delta }_A$ in the rational Chow ring of $X^n$ for $A$ ranging over $m$-element subsets of $\{1,\dots ,n\}$. 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 $e_k(x_{a_1},\dots ,x_{a_m})\in \mathbb{Z}[x_1,\dots ,x_n]$ are generated by the $S_n$-translates of a certain alternating sum over the facets of a hyperoctahedron.

8 pages, 1 figure, comments welcome!