Schubert puzzles and integrability I: Invariant trilinear forms

The puzzle rules for computing Schubert calculus on d d -step flag manifolds, proven in [Allen Knutson and Terence Tao, Puzzles and (equivariant) cohomology of Grassmannians , Duke Math. J. 119 (2003), pp. 221–260] for 1 1 -step, by Anders Skovsted Buch, Andrew Kresch, Kevin Purbhoo, and Harry Tamvakis [J. Algebraic Combin. 44 (2016), pp. 973–1007] for 2 2 -step, and conjectured by Izzet Coskun and Ravi Vakil [ Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus , Amer. Math. Soc., Providence, RI, 2009, pp. 77–124] for 3 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 R -matrices of those representations (which, for 2 2 -step flag manifolds, involve triality of D 4 D_4 ) degenerate to give us puzzle formulæ for two previously unsolved Schubert calculus problems: K T ( 2 K_T(2 -step flag manifolds ) ) and K ( 3 K(3 -step flag manifolds ) ) . The K ( 3 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 H^*(3 -step flag manifolds ) ) .