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<p>We show that <i>interlacing triangular arrays</i>, introduced by Aggarwal–Borodin–Wheeler to study certain probability measures, can be used to compute structure constants for multiplying Schubert classes in the <i>K</i>-theory of Grassmannians, in the cohomology of their cotangent bundles, and in the cohomology of partial flag varieties. Our results are achieved by establishing a splitting lemma, allowing for interlacing triangular arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 with certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson–Tao puzzles. Along the way, we prove one enumerative conjecture of Aggarwal–Borodin–Wheeler and disprove another.</p>

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Interlacing Triangles, Schubert Puzzles, and Graph Colorings

  • 期刊论文

  • 发表日期：

    2025年5月3日

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  • Christian Gaetz,
  • Yibo Gao

摘要

We show that interlacing triangular arrays, introduced by Aggarwal–Borodin–Wheeler to study certain probability measures, can be used to compute structure constants for multiplying Schubert classes in the K-theory of Grassmannians, in the cohomology of their cotangent bundles, and in the cohomology of partial flag varieties. Our results are achieved by establishing a splitting lemma, allowing for interlacing triangular arrays of high rank to be decomposed into arrays of lower rank, and by constructing a bijection between interlacing triangular arrays of rank 3 with certain proper vertex colorings of the triangular grid graph that factors through generalizations of Knutson–Tao puzzles. Along the way, we prove one enumerative conjecture of Aggarwal–Borodin–Wheeler and disprove another.