Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Lara Bossinger ; Fatemeh Mohammadi ; Alfredo Nájera Chávez

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gröbner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh–Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $\mathbb A^m \to X_C$. We apply this construction to the Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Plücker embeddings and the Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley–Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for ${\rm Gr}(2,\mathbb C^n)$ we show how Escobar–Harada's mutation of Newton–Okounkov bodies can be recovered as tropicalized cluster mutation.

Détail

Keywords: cluster algebras, Gröbner basis, Gröbner fan, Grassmannians, flat degenerations, Newton–Okounkov bodies.

Bibliographie

[1] Adams W. W., Loustaunau P., An introduction to Gröbner bases, Graduate Studies in Mathematics, 3, Amer. Math. Soc., Providence, RI, 1994 | DOI | MR | Zbl

[2] Akhtar M., Coates T., Galkin S., Kasprzyk A. M., “Minkowski polynomials and mutations”, SIGMA, 8 (2012), 094, 707 pp., arXiv: 1212.1785 | DOI | MR | Zbl

[3] Altmann K., Hausen J., “Polyhedral divisors and algebraic torus actions”, Math. Ann., 334 (2006), 557–607, arXiv: math.AG/0306285 | DOI | MR | Zbl

[4] Bossinger L., Grassmannians and universal coefficients for cluster algebras: computational data for ${\rm Gr}(3,6)$, https://www.matem.unam.mx/l̃ara/clusterGr36

[5] Bossinger L., “Full-rank valuations and toric initial ideals”, Int. Math. Res. Not., 2021 (2021), 7715–7763, arXiv: 1903.11068 | DOI | MR

[6] Bossinger L., Cheung M. W., Magee T., Nájera Chávez A., On cluster duality for Grassmannians, in preparation

[7] Bossinger L., Fang X., Fourier G., Hering M., Lanini M., “Toric degenerations of ${\rm Gr}(2, n)$ and ${\rm Gr}(3, 6)$ via plabic graphs”, Ann. Comb., 22 (2018), 491–512, arXiv: 1612.03838 | DOI | MR | Zbl

[8] Bossinger L., Fourier G., “String cone and superpotential combinatorics for flag and Schubert varieties in type A”, J. Combin. Theory Ser. A, 167 (2019), 213–256, arXiv: 1611.06504 | DOI | MR | Zbl

[9] Bossinger L., Frías-Medina B., Magee T., Nájera Chávez A., “Toric degenerations of cluster varieties and cluster duality”, Compos. Math., 156 (2020), 2149–2206, arXiv: 1809.08369 | DOI | MR | Zbl

[10] Bossinger L., Harada M., Mohammadi F., Toric degenerations of tropical Grassmannians, Extended abstract in Mathematisches Forschungsinstitut Oberwolfach, Workshop Report No 43, 2019 | Zbl

[11] Buchberger B., “An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal”, J. Symbolic Comput., 41 (2006), 475–511 | DOI | MR | Zbl

[12] Clarke O., Higashitani A., Mohammadi F., “Combinatorial mutations and block diagonal polytopes”, Collect. Math. (to appear) , arXiv: 2010.04079 | DOI

[13] Clarke O., Mohammadi F., “Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux”, J. Algebra, 559 (2020), 646–678, arXiv: 1904.00981 | DOI | MR | Zbl

[14] Cox D. A., “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom., 4 (1995), 17–50, arXiv: alg-geom/9210008 | MR | Zbl

[15] Cox D. A., Little J. B., Schenck H. K., Toric varieties, Graduate Studies in Mathematics, 124, Amer. Math. Soc., Providence, RI, 2011 | DOI | MR | Zbl

[16] Derksen H., Weyman J., Zelevinsky A., “Quivers with potentials and their representations II: applications to cluster algebras”, J. Amer. Math. Soc., 23 (2010), 749–790, arXiv: 0904.0676 | DOI | MR | Zbl

[17] Dolgachev I., “Weighted projective varieties”, Group Actions and Vector Fields (Vancouver, BC, 1981), Lecture Notes in Math., 956, Springer, Berlin, 1982, 34–71 | DOI | MR

[18] Eisenbud D., Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 | DOI | MR | Zbl

[19] Escobar L., Harada M., “Wall-crossing for Newton–Okounkov bodies and the tropical Grassmannian”, Int. Math. Res. Not. (to appear) , arXiv: 1912.04809 | DOI

[20] Fock V. V., Goncharov A. B., “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 865–930, arXiv: math.AG/0311245 | DOI | MR | Zbl

[21] Fomin S., Shapiro M., Thurston D., “Cluster algebras and triangulated surfaces. I Cluster complexes”, Acta Math., 201 (2008), 83–146, arXiv: math.RA/0608367 | DOI | MR | Zbl

[22] Fomin S., Thurston D., “Cluster algebras and triangulated surfaces. Part II Lambda lengths”, Mem. Amer. Math. Soc., 255, 2018, v+97 pp., arXiv: 1210.5569 | DOI | MR

[23] Fomin S., Zelevinsky A., “Cluster algebras. I Foundations”, J. Amer. Math. Soc., 15 (2002), 497–529, arXiv: math.RT/0104151 | DOI | MR | Zbl

[24] Fomin S., Zelevinsky A., “$Y$-systems and generalized associahedra”, Ann. of Math., 158 (2003), 977–1018, arXiv: hep-th/0111053 | DOI | MR | Zbl

[25] Fomin S., Zelevinsky A., “Cluster algebras. IV Coefficients”, Compos. Math., 143 (2007), 112–164, arXiv: math.RA/0602259 | DOI | MR | Zbl

[26] Fujita N., Oya H., Newton–Okounkov polytopes of Schubert varieties arising from cluster structures, arXiv: 2002.09912

[27] Grayson D., Stillman M., Macaulay2, a software system for research in algebraic geometry, https://faculty.math.illinois.edu/Macaulay2/

[28] Gross M., Hacking P., Keel S., Kontsevich M., “Canonical bases for cluster algebras”, J. Amer. Math. Soc., 31 (2018), 497–608, arXiv: 1411.1394 | DOI | MR | Zbl

[29] Herzog J., Hibi T., Monomial ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011 | DOI | MR | Zbl

[30] Hosgood T., An introduction to varieties in weighted projective space, arXiv: 1604.02441

[31] Ilten N. O., “Mutations of Laurent polynomials and flat families with toric fibers”, SIGMA, 8 (2012), 047, 7 pp., arXiv: 1205.4664 | DOI | MR | Zbl

[32] Kateri M., Mohammadi F., Sturmfels B., “A family of quasisymmetry models”, J. Algebr. Stat., 6 (2015), 1–16, arXiv: 1403.0547 | DOI | MR | Zbl

[33] Kaveh K., Manon C., Toric flat families, valuations, and tropical geometry over the semifield of piecewise linear functions, arXiv: 1907.00543

[34] Kaveh K., Manon C., “Khovanskii bases, higher rank valuations, and tropical geometry”, SIAM J. Appl. Algebra Geom., 3 (2019), 292–336, arXiv: 1610.00298 | DOI | MR | Zbl

[35] Keller B., Quiver mutation in JavaScript, https://webusers.imj-prg.fr/b̃ernhard.keller/quivermutation/

[36] Keller B., “Cluster algebras and derived categories”, Derived Categories in Algebraic Geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, 123–183, arXiv: 1202.4161 | DOI | MR | Zbl

[37] Kreuzer M., Robbiano L., Computational commutative algebra, v. 2, Springer-Verlag, Berlin, 2005 | DOI | MR | Zbl

[38] Maclagan D., Sturmfels B., Introduction to tropical geometry, Graduate Studies in Mathematics, 161, Amer. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

[39] Mohammadi F., Shaw K., “Toric degenerations of Grassmannians from matching fields”, Algebr. Comb., 2 (2019), 1109–1124, arXiv: 1809.01026 | DOI | MR | Zbl

[40] Mora T., Robbiano L., “The Gröbner fan of an ideal”, J. Symbolic Comput., 6 (1988), 183–208 | DOI | MR | Zbl

[41] Muller G., Rajchgot J., Zykoski B., “Lower bound cluster algebras: presentations, Cohen–Macaulayness, and normality”, Algebr. Comb., 1 (2018), 95–114, arXiv: 1508.02314 | DOI | MR | Zbl

[42] Nájera Chávez A., “A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients”, Math. Z., 291 (2019), 1495–1523, arXiv: 1512.07939 | DOI | MR | Zbl

[43] Reading N., “Universal geometric cluster algebras”, Math. Z., 277 (2014), 499–547, arXiv: 1209.3987 | DOI | MR | Zbl

[44] Reading N., “Universal geometric cluster algebras from surfaces”, Trans. Amer. Math. Soc., 366 (2014), 6647–6685, arXiv: 1209.4095 | DOI | MR | Zbl

[45] Reid M., Graded rings and varieties in weighted projective space, https://homepages.warwick.ac.uk/m̃asda/surf/more/grad.pdf

[46] Rietsch K., Williams L., “Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians”, Duke Math. J., 168 (2019), 3437–3527, arXiv: 1712.00447 | DOI | MR | Zbl

[47] Scott J. S., “Grassmannians and cluster algebras”, Proc. London Math. Soc., 92 (2006), 345–380, arXiv: math.CO/0311148 | DOI | MR | Zbl

[48] Serhiyenko K., Sherman-Bennett M., Williams L., “Cluster structures in Schubert varieties in the Grassmannian”, Proc. London Math. Soc., 119 (2019), 1694–1744, arXiv: 1902.00807 | DOI | MR | Zbl

[49] Speyer D., Sturmfels B., “The tropical Grassmannian”, Adv. Geom., 4 (2004), 389–411, arXiv: math.AG/0304218 | DOI | MR | Zbl

[50] Speyer D., Williams L., “The tropical totally positive Grassmannian”, J. Algebraic Combin., 22 (2005), 189–210, arXiv: math.CO/0312297 | DOI | MR | Zbl

[51] Sturmfels B., Gröbner bases and convex polytopes, University Lecture Series, 8, Amer. Math. Soc., Providence, RI, 1996 | DOI | MR | Zbl

[52] Usnich A., On the Cremona group and its subgroups, Ph.D. Thesis, Université Pierre et Marie Curie - Paris VI, 2008 | MR