Algebraic and enumerative combinatorics seminar-Ashleigh Adams-Promotion, plane partitions, partial evaluations, and webs

Abstract: Webs are graphical objects that give a tangible, combinatorial way to compute and classify tensor invariants. Recently, Gaetz, Pechenik, Pfannerer, Striker, and Swanson (arXiv:2306.12501) found a rotation-invariant web basis for SL₄, as well as its quantum deformation U_q(sl₄), and a bijection between move equivalence classes of SL₄-webs and fluctuating tableaux such that web rotation corresponds to tableau promotion. They also found a bijection between the set of plane partitions in an a×b×c box and a benzene move equivalence class of SL₄-webs by determining the corresponding oscillating tableau. In this talk, I will similarly find the oscillating tableaux corresponding to plane partitions in certain symmetry classes by characterizing them via certain lattice words. A dynamical action on tableaux, called promotion, corresponds to rotation of SL₄-webs. I will show how promotion of certain subtableaux aligns with rotation of their respective webs. I will also show that this correspondence maps through a projection to either SL₂ or SL₃ webs. Moreover, this projection is exactly a partial evaluation of webs. This talk will be given through the lens of the combinatorics of webs and tableaux. Some of this work is joint with Jessica Striker.