Proceedings of the American Mathematical Society Series B· 2026Q1
Extending the 𝐚𝐛-index
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- 2026year
Short summary
The Poincaré-extended ab-index for finite, graded, bounded posets is obtained from the ab-index via the ω-transformation, confirming a conjecture.
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Key points
- The ω-transformation connects the ab-index to the Poincaré-extended ab-index for specific posets.
- This result confirms a conjecture by Dorpalen-Barry, Maglione, and Stump.
- The ω-transformation generalizes existing concepts like the (c-2d)-index and Chow ring decomposition.
- The approach is based on a known identity within the incidence algebra.
AI-generated from the title and abstract; the full text is not read.
Abstract
We prove for finite, graded, bounded posets, that the Poincaré-extended a b \mathbf {a}\mathbf {b} -index is obtained from the a b \mathbf {a}\mathbf {b} -index via the ω \omega -transformation as conjectured by Dorpalen-Barry, Maglione, and the second author. This ω \omega -transformation is a generalization of the ( c (\mathbf {c} - 2 d ) 2\mathbf {d}) -index, and it also generalizes the numerical canonical decomposition of the Chow ring. It provides a conceptually new approach to extended a b \mathbf {a}\mathbf {b} -indices and Chow polynomials, beyond R R -labeled posets, based on a well-known identity in the incidence algebra.
The authors' abstract, as published at the source. Proceedings of the American Mathematical Society Series B, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics