In 1997, Terry Tao and I invented "puzzles" to study Horn's problem, slightly after Klyachko had used Schubert calculus on Grassmannians for the same purpose. It turns out that these puzzles, with their three edge labels, connect more directly to Schubert calculus. Last year, Pechenik-Yong and Wheeler--Zinn-Justin used them to compute equivariant K-theory of Grassmannians (using some new pieces). But what about d-step flag manifolds? In 2014, (equivariant) cohomology of 2-step flag manifolds was puzzlified, proving a conjecture of mine from 1999.

To force the three sides of a puzzle to relate to the same flag manifold, we assign a vector to each edge label, which leads to vector configurations of 3 vectors (for d=1), 8 for d=2, and 27 for d=3; the configurations correspond to the weights of the minuscule representations of A_2, D_4, and E_6. From the Jimbo R-matrices of these representations, we derive some new puzzle rules. Speaker(s): Allen Knutson (Cornell)