$$
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$$
Symbol map and intertwining
The symbol map \(\sigma_k\) extracts from a tangent cube E0016
a symmetric probe E0027 that carries the same differential
information. It intertwines the cubical and symmetric
pushforwards.
(1) Definition (Symbol map).
\[
\sigma_k: CT_k(X; x) \to ST_k(X; x),
\qquad
\sigma_k(c) := \sum_{\pi \in \Part(k)}
\prod_{A \in \pi} c(A),
\]
where the products are in the symmetric algebra. On affine
cubes, \(\sigma_k(\Aff(v_1, \dots, v_k)) = v_1 \cdots v_k\).
(2) Lemma (Spanning). For every \(1 \leq r \leq k\) and
\(v_1, \dots, v_r \in E\), the monomial \(v_1 \cdots v_r\) lies
in the image of \(\sigma_k\). The image of \(\sigma_k\) together
with \(1\) spans \(ST_k(X; x)\).
Proof.
Choose a partition \([k] = B_1 \sqcup \cdots \sqcup B_r\) and
set \(c(B_j) := v_j\), \(c(A) := 0\) for all other \(A\). Only
\(\pi = \set{B_1, \dots, B_r}\) contributes to \(\sigma_k(c)\),
giving \(v_1 \cdots v_r\).
Proof.
The collapse partition formula E0031
\(D^\square(f; x; c) =
\sum_\pi D(f; x; (c(A))_{A \in \pi})\) is exactly the
differential pairing evaluated at \(\sigma_k(c)\).
Proof.
Expand \(\sigma_k(D^\square_+(\phi; x) c)\): the partition
formula E0031 on each face gives
\(\sum_{\pi \in \Part(k)} \prod_{A \in \pi}
\sum_{\rho_A \in \Part(A)} D(\phi; x; (c(B))_{B \in \rho_A})\).
Expanding the product, the index data \((\pi, (\rho_A))\)
correspond bijectively to pairs \((\rho, \KQ)\) of a partition
\(\rho \in \Part(k)\) and a partition \(\KQ\) of the block set of
\(\rho\): set \(\rho = \bigsqcup_A \rho_A\) and
\(\KQ = \set{\rho_A : A \in \pi}\).
On the other side,
\(D_+(\phi; x; \sigma_k(c)) =
\sum_{\rho \in \Part(k)} \sum_{\KQ \in \Part(\rho)}
\prod_{Q \in \KQ} D(\phi; x; (c(B))_{B \in Q})\).
The two sums agree term by term.
Used by: