Schubert puzzles and integrability I: invariant trilinear forms

<p> The <italic>puzzle rules</italic> for computing Schubert calculus on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step flag manifolds, proven in [Allen Knutson and Terence Tao, <italic>Puzzles and (equivariant) cohomology of Grassmannians</italic> , Duke Math. J. 119 (2003), pp. 221–260] for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step, by Anders Skovsted Buch, Andrew Kresch, Kevin Purbhoo, and Harry Tamvakis [J. Algebraic Combin. 44 (2016), pp. 973–1007] for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step, and conjectured by Izzet Coskun and Ravi Vakil [ <italic>Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus</italic> , Amer. Math. Soc., Providence, RI, 2009, pp. 77–124] for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step, lead to vector configurations (one vector for each puzzle edge label) that we recognize as the weights of some minuscule representations. The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -matrices of those representations (which, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step flag manifolds, involve triality of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D 4"> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">D_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) degenerate to give us puzzle formulæ for two previously unsolved Schubert calculus problems: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript upper T Baseline left-parenthesis 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K_T(2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step flag manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right-parenthesis"> <mml:semantics> <mml:mo stretchy="false">)</mml:mo> <mml:annotation encoding="application/x-tex">)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K(3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step flag manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right-parenthesis"> <mml:semantics> <mml:mo stretchy="false">)</mml:mo> <mml:annotation encoding="application/x-tex">)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K(3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step flag manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right-parenthesis"> <mml:semantics> <mml:mo stretchy="false">)</mml:mo> <mml:annotation encoding="application/x-tex">)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> formula, which involves 151 new puzzle pieces, implies Buch’s correction to the first author’s 1999 conjecture for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis 3"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">H^*(3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -step flag manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="right-parenthesis"> <mml:semantics> <mml:mo stretchy="false">)</mml:mo> <mml:annotation encoding="application/x-tex">)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p>