### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 078, 34 pages      arXiv:1704.00839      https://doi.org/10.3842/SIGMA.2018.078

### $t$-Unique Reductions for Mészáros's Subdivision Algebra

Darij Grinberg
School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA

Received November 22, 2017, in final form July 15, 2018; Published online July 26, 2018

Abstract
Fix a commutative ring $\mathbf{k}$, two elements $\beta \in\mathbf{k}$ and $\alpha\in\mathbf{k}$ and a positive integer $n$. Let $\mathcal{X}$ be the polynomial ring over $\mathbf{k}$ in the $n(n-1) /2$ indeterminates $x_{i,j}$ for all $1\leq i$<$j\leq n$. Consider the ideal $\mathcal{J}$ of $\mathcal{X}$ generated by all polynomials of the form $x_{i,j}x_{j,k}-x_{i,k} ( x_{i,j}+x_{j,k}+\beta ) -\alpha$ for $1\leq i$<$j$<$k\leq n$. The quotient algebra $\mathcal{X}/\mathcal{J}$ (at least for a certain choice of $\mathbf{k}$, $\beta$ and $\alpha$) 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 $\mathcal{X}$ is said to be pathless if it has no divisors of the form $x_{i,j}x_{j,k}$ with $1\leq i$<$j$<$k\leq n$. The residue classes of these pathless monomials span the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, but (in general) are $\mathbf{k}$-linearly dependent. More combinatorially: reducing a given $p\in\mathcal{X}$ modulo the ideal $\mathcal{J}$ by applying replacements of the form $x_{i,j}x_{j,k}\mapsto x_{i,k} ( x_{i,j}+x_{j,k}+\beta ) +\alpha$ always eventually leads to a $\mathbf{k}$-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 $\mathbf{k}$-algebra homomorphism $D$ from $\mathcal{X}$ into the polynomial ring $\mathbf{k} [ t_{1},t_{2},\ldots,t_{n-1} ]$ that sends each $x_{i,j}$ to $t_{i}$. We show the following fact (generalizing a conjecture of Mészáros): If $p\in\mathcal{X}$, and if $q\in\mathcal{X}$ is a $\mathbf{k}$-linear combination of pathless monomials satisfying $p\equiv q\operatorname{mod}\mathcal{J}$, then $D(q)$ does not depend on $q$ (as long as $\beta$, $\alpha$ and $p$ are fixed). Thus, the above way of reducing a $p\in\mathcal{X}$ modulo $\mathcal{J}$ may lead to different results, but all of them become identical once $D$ is applied. We also find an actual basis of the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, using what we call forkless monomials.

Key words: subdivision algebra; Yang-Baxter relations; Gröbner bases; Arnold relations; Orlik-Terao algebras; noncommutative algebra.

pdf (676 kb)   tex (41 kb)

References

1. Aparicio Monforte A., Kauers M., Formal Laurent series in several variables, Expo. Math. 31 (2013), 350-367.
2. Arnold V.I., The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138-140.
3. Becker T., Weispfenning V., Gröbner bases: a computational approach to commutative algebra, Graduate Texts in Mathematics, Vol. 141, Springer-Verlag, New York, 1993.
4. Cordovil R., Etienne G., A note on the Orlik-Solomon algebra, European J. Combin. 22 (2001), 165-170, math.CO/0203152.
5. Cox D.A., Little J., O'Shea D., Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, 4th ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015.
6. de Graaf W., Computational algebra, Lecture notes, Version September 21, 2016, available at http://www.science.unitn.it/~degraaf/compalg/notes.pdf.
7. Escobar L., Mészáros K., Subword complexes via triangulations of root polytopes, Algebraic Combin. 1 (2018), 395-414, arXiv:1502.03997.
8. Grinberg D., $t$-unique reductions for Mészáros's subdivision algebra, detailed version of the present paper, arXiv:1704.00839v6.
9. Grinberg D., Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?, mathOverflow post \#286510, available at https://mathoverflow.net/questions/286510.
10. Guo L., What is $\ldots$ a Rota-Baxter algebra?, Notices Amer. Math. Soc. 56 (2009), 1436-1437.
11. Horiuchi H., Terao H., The Poincaré series of the algebra of rational functions which are regular outside hyperplanes, J. Algebra 266 (2003), 169-179, math.CO/0202296.
12. Kirillov A.N., On some quadratic algebras, q-alg/9705003.
13. Kirillov A.N., On some algebraic and combinatorial properties of Dunkl elements, Internat. J. Modern Phys. B 26 (2012), 1243012, 28 pages.
14. Kirillov A.N., On some quadratic algebras I $\frac{1}{2}$: combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA 12 (2016), 002, 172 pages, arXiv:1502.00426.
15. Mathieu O., The symplectic operad, in Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993), Progr. Math., Vol. 131, Editors S. Gindikin, J. Lepowsky, R.L. Wilson, Birkhäuser Boston, Boston, MA, 1995, 223-243.
16. McBreen M., Proudfoot N., Intersection cohomology and quantum cohomology of conical symplectic resolutions, Algebr. Geom. 2 (2015), 623-641, arXiv:1410.6240.
17. Mészáros K., Root polytopes, triangulations, and the subdivision algebra. I, Trans. Amer. Math. Soc. 363 (2011), 4359-4382, arXiv:0904.2194.
18. Mészáros K., St. Dizier A., From generalized permutahedra to Grothendieck polynomials via flow polytopes, arXiv:1705.02418.
19. Monasse D., Introduction aux bases de Gröbner: théorie et pratique, Lecture notes, Version November 19, 2002, available at http://denis.monasse.free.fr/denis/articles/grobner.pdf.
20. Moseley D., Equivariant cohomology and the Varchenko-Gelfand filtration, J. Algebra 472 (2017), 95-114, arXiv:1110.5369.
21. Orlik P., Terao H., Commutative algebras for arrangements, Nagoya Math. J. 134 (1994), 65-73.
22. Proudfoot N., Speyer D., A broken circuit ring, Beiträge Algebra Geom. 47 (2006), 161-166, math.CO/0410069.
23. SageMath, the Sage Mathematics Software System, Version 7.6, 2017, available at http://www.sagemath.org.
24. Schenck H., Tohaneanu Ş.O., The Orlik-Terao algebra and 2-formality, Math. Res. Lett. 16 (2009), 171-182, arXiv:0901.0253.
25. Stanley R.P., Enumerative combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, 2nd ed., Cambridge University Press, Cambridge, 2012, available at http://math.mit.edu/~rstan/ec/.
26. Stanley R.P., Catalan numbers, Cambridge University Press, New York, 2015.
27. Sturmfels B., Algorithms in invariant theory, 2nd ed., Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 2008.
28. Terao H., Algebras generated by reciprocals of linear forms, J. Algebra 250 (2002), 549-558.
29. Varchenko A.N., Gel'fand I.M., Heaviside functions of a configuration of hyperplanes, Funct. Anal. Appl. 21 (1987), 255-270.
30. Yuzvinsky S., Orlik-Solomon algebras in algebra and topology, Russian Math. Surveys 56 (2001), 293-364.