Interlacing triangles, Schubert puzzles, and graph colorings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912359434420224 |
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| author | Gaetz, Christian Gao, Yibo |
| author_facet | Gaetz, Christian Gao, Yibo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interlacing triangles, Schubert puzzles, and graph colorings Gaetz, Christian Gao, Yibo Combinatorics Algebraic Geometry Probability 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. |
| title | Interlacing triangles, Schubert puzzles, and graph colorings |
| topic | Combinatorics Algebraic Geometry Probability |
| url | https://arxiv.org/abs/2408.07863 |