An interactive example of a theorem from the upcoming paper "Promotion and projections of SL_r-invariants" (2025+), Ashleigh Adams and Jessica Striker.
Instructions: Drag the big web (on the left) to rotate both webs. (This feature currently does not work on phones.)
Description: The SL_4-web on the left (with separation labels for edges incident to vertices on the boundary given in orange and with bace face F_0) is in bijection with the fundamental domain of a totally symmetric self-complementary plane partition.
One can project the web on the left to the SL_2-web on the right (Adams, Striker 2025+).
Underneath both webs is the lattice word of the web.
By rotating the webs, the location of the base faces change, thereby changing the corresponding lattice words.
Each lattice word is in bijection with an oscillating tableau.
This example demonstrates how rotation on both webs and the dynamical action of promotion on both lattice words align through a projection found in "Webification of symmetry classes of plane partitions" (Adams, Sriker 2025+).
\[
\mathsf{pro}^k(\omega) \quad \text{as } k \text{ increases, with } \omega = 1112\overline{4}2\overline{4}2(34)44(34)(34)
\]