t-Unique Reductions for Mészáros's Subdivision Algebra
Fix a commutative ring k, two elements β ∈ k and α ∈ k, and a positive integer n. Let X be the polynomial ring over k in the n(n−1)/2 indeterminates xᵢ,ⱼ for all 1 ≤ i < j ≤ n. Consider the ideal J of X generated by all polynomials of the form xᵢ,ⱼxⱼ,ₖ−xᵢ,ₖ(xᵢ,ⱼ+xⱼ,ₖ+β)−α for 1 ≤ i< j < k ≤...
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| Date: | 2018 |
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Інститут математики НАН України
2018
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| Cite this: | t-Unique Reductions for Mészáros's Subdivision Algebra / D. Grinberg // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 30 назв. — англ. |
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Grinberg, D. 2025-11-26T11:33:42Z 2018 t-Unique Reductions for Mészáros's Subdivision Algebra / D. Grinberg // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 30 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E15; 05E40 arXiv: 1704.00839 https://nasplib.isofts.kiev.ua/handle/123456789/209772 https://doi.org/10.3842/SIGMA.2018.078 Fix a commutative ring k, two elements β ∈ k and α ∈ k, and a positive integer n. Let X be the polynomial ring over k in the n(n−1)/2 indeterminates xᵢ,ⱼ for all 1 ≤ i < j ≤ n. Consider the ideal J of X generated by all polynomials of the form xᵢ,ⱼxⱼ,ₖ−xᵢ,ₖ(xᵢ,ⱼ+xⱼ,ₖ+β)−α for 1 ≤ i< j < k ≤ n. The quotient algebra X/J (at least for a certain choice of k, β, and α) has been introduced by Karola Mészáros in [Trans. Amer. Math. Soc. 363 (2011), 4359-4382] as a commutative analogue of Anatol Kirillov's quasi-classical Yang-Baxter algebra. A monomial in X is said to be pathless if it has no divisors of the form xᵢ,ⱼxⱼ,ₖ with 1 ≤ i < j < k ≤n. The residue classes of these pathless monomials span the k-module X/J, but (in general) are k-linearly dependent. More combinatorially: reducing a given p∈X modulo the ideal J by applying replacements of the form xᵢ,ⱼxⱼ,ₖ↦xᵢ,ₖ(xᵢ,ⱼ+xⱼ,ₖ+β)+α always eventually leads to a k-linear combination of pathless monomials, but the result may depend on the choices made in the process. More recently, the study of Grothendieck polynomials has led Laura Escobar and Karola Mészáros [Algebraic Combin. 1 (2018), 395-414] to defining a k-algebra homomorphism D from X into the polynomial ring k[t₁,t₂,…,tₙ₋₁] that sends each xᵢ,ⱼ to tᵢ. We show the following fact (generalizing a conjecture of Mészáros): If p ∈ X, and if q ∈ X is a k-linear combination of pathless monomials satisfying p ≡ q mod J, then D(q) does not depend on q (as long as β, α, and p are fixed). Thus, the above way of reducing a p ∈ X modulo J may lead to different results, but all of them become identical once D is applied. We also find an actual basis of the k-module X/J, using what we call forkless monomials. The SageMath computer algebra system [23] was of great service during the development of the results below. Conversations with Nick Early have led me to the ideas in Section 5.3, and Victor Reiner has helped me concretize them. This paper has furthermore profited from enlightening comments by Ricky Liu, Karola Mészáros, Nicholas Proudfoot, Travis Scrimshaw, Richard Stanley, the anonymous referees, and the editor. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications t-Unique Reductions for Mészáros's Subdivision Algebra Article published earlier |
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| title |
t-Unique Reductions for Mészáros's Subdivision Algebra |
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t-Unique Reductions for Mészáros's Subdivision Algebra Grinberg, D. |
| title_short |
t-Unique Reductions for Mészáros's Subdivision Algebra |
| title_full |
t-Unique Reductions for Mészáros's Subdivision Algebra |
| title_fullStr |
t-Unique Reductions for Mészáros's Subdivision Algebra |
| title_full_unstemmed |
t-Unique Reductions for Mészáros's Subdivision Algebra |
| title_sort |
t-unique reductions for mészáros's subdivision algebra |
| author |
Grinberg, D. |
| author_facet |
Grinberg, D. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Fix a commutative ring k, two elements β ∈ k and α ∈ k, and a positive integer n. Let X be the polynomial ring over k in the n(n−1)/2 indeterminates xᵢ,ⱼ for all 1 ≤ i < j ≤ n. Consider the ideal J of X generated by all polynomials of the form xᵢ,ⱼxⱼ,ₖ−xᵢ,ₖ(xᵢ,ⱼ+xⱼ,ₖ+β)−α for 1 ≤ i< j < k ≤ n. The quotient algebra X/J (at least for a certain choice of k, β, and α) has been introduced by Karola Mészáros in [Trans. Amer. Math. Soc. 363 (2011), 4359-4382] as a commutative analogue of Anatol Kirillov's quasi-classical Yang-Baxter algebra. A monomial in X is said to be pathless if it has no divisors of the form xᵢ,ⱼxⱼ,ₖ with 1 ≤ i < j < k ≤n. The residue classes of these pathless monomials span the k-module X/J, but (in general) are k-linearly dependent. More combinatorially: reducing a given p∈X modulo the ideal J by applying replacements of the form xᵢ,ⱼxⱼ,ₖ↦xᵢ,ₖ(xᵢ,ⱼ+xⱼ,ₖ+β)+α always eventually leads to a k-linear combination of pathless monomials, but the result may depend on the choices made in the process. More recently, the study of Grothendieck polynomials has led Laura Escobar and Karola Mészáros [Algebraic Combin. 1 (2018), 395-414] to defining a k-algebra homomorphism D from X into the polynomial ring k[t₁,t₂,…,tₙ₋₁] that sends each xᵢ,ⱼ to tᵢ. We show the following fact (generalizing a conjecture of Mészáros): If p ∈ X, and if q ∈ X is a k-linear combination of pathless monomials satisfying p ≡ q mod J, then D(q) does not depend on q (as long as β, α, and p are fixed). Thus, the above way of reducing a p ∈ X modulo J may lead to different results, but all of them become identical once D is applied. We also find an actual basis of the k-module X/J, using what we call forkless monomials.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209772 |
| citation_txt |
t-Unique Reductions for Mészáros's Subdivision Algebra / D. Grinberg // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 30 назв. — англ. |
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2025-12-07T12:59:51Z |
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2025-12-07T12:59:51Z |
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1850885993250422784 |