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Schubert puzzles and integrability I: invariant trilinear forms

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arxiv 1706.10019 v8 pith:D5Y4OPO7 submitted 2017-06-30 math.CO math-phmath.AGmath.MP

Allen Knutson , Paul Zinn-Justin This is my paper · ORCID
classification math.COmath-phmath.AGmath.MP
keywords stepflagmanifoldspuzzleschubertbuchcalculusrepresentations
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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$)$.

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Cited by 3 Pith papers

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    Scaled random hives with GUE boundary conditions converge in probability to a unique continuum hive whose value at a point v is the supremum of a functional over asymptotic height functions of lozenge tilings.

  2. Positivity of Schubert Coefficients

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    Under GRH and the Miltersen-Vinodchandran assumption, the positivity of Schubert coefficients has a positive rule, equivalent to the problem being in NP.

  3. Introduction to the Cohomology of the Flag Variety

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    A survey chapter presenting the cohomology of flag varieties and Schubert and Schur polynomials as the rigorous basis for solving Schubert's enumerative geometry problems.

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