Schubert puzzles and integrability I: invariant trilinear forms

Abstract

The puzzle rules for computing Schubert calculus on $d$-step flag manifolds, proven in [Knutson Tao 2003] for $1$-step, in [Buch Kresch Purbhoo Tamvakis 2016] for $2$-step, and conjectured in [Coskun Vakil 2009] for $3$-step, lead to vector configurations (one vector for each puzzle edge label) that we recognize as the weights of some minuscule representations. The $R$-matrices of those representations (which, for $2$-step flag manifolds, involve triality of $D_4$) degenerate to give us puzzle formulae for two previously unsolved Schubert calculus problems: $K_T(2$-step flag manifolds$)$ and $K(3$-step flag manifolds$)$. The $K(3$-step flag manifolds$)$ formula, which involves 151 new puzzle pieces, implies Buch's correction to the first author's 1999 conjecture for $H^*(3$-step flag manifolds$)$.