Factorizations by column blocks and Schubert polynomials as products of key polynomials
Abstract
We classify the Schubert polynomials that are products of key polynomials and explicitly determine their finest key factorization. Their indexing permutations are precisely those avoiding ten classical patterns of length six, which form the minimal pattern basis. More generally, the connected components of the column interaction graph give the unique finest factorization of a dual flagged Weyl character into nonconstant diagram characters. Every other such factorization is obtained by grouping these components. We also prove Makhija's conjecture on products of elementary symmetric polynomials in initial alphabets, allowing repeated alphabets. In this case, the elementary factors are uniquely determined, up to order, by the nonempty Rothe columns.
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BibTeX
@misc{han2026factorizations,
title = {{Factorizations by column blocks and Schubert polynomials as products of key polynomials}},
author = {Xuanying Han and Shunli Ma and Sophie C. C. Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36993},
eprint = {2610.36993},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36993},
}
RIS
TY - UNPB
TI - Factorizations by column blocks and Schubert polynomials as products of key polynomials
AU - Xuanying Han
AU - Shunli Ma
AU - Sophie C. C. Sun
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36993
UR - https://arxiv.org/abs/2610.36993
AB - We classify the Schubert polynomials that are products of key polynomials and explicitly determine their finest key factorization. Their indexing permutations are precisely those avoiding ten classical patterns of length six, which form the minimal pattern basis. More generally, the connected components of the column interaction graph give the unique finest factorization of a dual flagged Weyl character into nonconstant diagram characters. Every other such factorization is obtained by grouping these components. We also prove Makhija's conjecture on products of elementary symmetric polynomials in initial alphabets, allowing repeated alphabets. In this case, the elementary factors are uniquely determined, up to order, by the nonempty Rothe columns.
ER -