Combinatorial mutations and block diagonal polytopes. (English) Zbl 1493.14071
A combinatorial mutation, introduced in [M. Akhtar et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 8, Paper 094, 17 p. (2012; Zbl 1280.52014)], is a transformation of the Newton polytope of a Laurent polynomial undergoing mutation; which can be considered as a local transformation for lattice polytopes. The theory of combinatorial mutations was further developed by Higashitani from a combinatorial viewpoint, and later has been used to study combinatorial mutation equivalence classes of Newton-Okounkov bodies of flag varieties [N. Fujita and A. Higashitani, Int. Math. Res. Not. 2021, No. 12, 9567–9607 (2021; Zbl 1524.14109)].
For the Grassmannian \(\text{Gr}(k, n)\), a matching field is a map taking each Plücker variable to a permutation, and can be interpreted as a choice of initial term for the corresponding Plücker form. They were introduced by B. Sturmfels and A. V. Zelevinsky [Adv. Math. 98, No. 1, 65–112 (1993; Zbl 0776.13009)] to study the Newton polytope of a product of maximal minors of a generic matrix and have proved to be a useful tool in many contexts.
In this paper, the authors use combinatorial mutations to find relations between matching field polytopes. In fact, each matching field \(\Lambda\) admits a toric ideal \(J_\Lambda\) with associated polytope \(P_\Lambda\), and they show that understanding the polytope associated to a matching field is equivalent to finding toric degenerations of the Grassmannian as the following:
Theorem 1. Let \(\Lambda\) be a coherent matching field for the Grassmannian \(\text{Gr}(k, n)\) with polytope \(P_\Lambda\). If \(P_\Lambda\) is obtained from the Gelfand-Tsetlin polytope by a sequence of combinatorial mutations, then \(\Lambda\) gives rise to a toric degeneration of \(\text{Gr}(k, n)\).
In particular, this paper investigate the block diagonal matching fields which are examples of coherent matching fields with particularly simple description. It shows that all block diagonal matching field polytopes are related by a sequence of combinatorial mutations:
Theorem 2. Any pair of block diagonal matching field polytopes can be obtained from one another by a sequence of combinatorial mutations such that all intermediate polytopes are matching field polytopes.
The matching fields associated to the intermediate polytopes can be thought of as interpolating between the block diagonal matching fields. As a result, the authors obtain a large family of toric degenerations for the Grassmannian given by matching fields.
For the Grassmannian \(\text{Gr}(k, n)\), a matching field is a map taking each Plücker variable to a permutation, and can be interpreted as a choice of initial term for the corresponding Plücker form. They were introduced by B. Sturmfels and A. V. Zelevinsky [Adv. Math. 98, No. 1, 65–112 (1993; Zbl 0776.13009)] to study the Newton polytope of a product of maximal minors of a generic matrix and have proved to be a useful tool in many contexts.
In this paper, the authors use combinatorial mutations to find relations between matching field polytopes. In fact, each matching field \(\Lambda\) admits a toric ideal \(J_\Lambda\) with associated polytope \(P_\Lambda\), and they show that understanding the polytope associated to a matching field is equivalent to finding toric degenerations of the Grassmannian as the following:
Theorem 1. Let \(\Lambda\) be a coherent matching field for the Grassmannian \(\text{Gr}(k, n)\) with polytope \(P_\Lambda\). If \(P_\Lambda\) is obtained from the Gelfand-Tsetlin polytope by a sequence of combinatorial mutations, then \(\Lambda\) gives rise to a toric degeneration of \(\text{Gr}(k, n)\).
In particular, this paper investigate the block diagonal matching fields which are examples of coherent matching fields with particularly simple description. It shows that all block diagonal matching field polytopes are related by a sequence of combinatorial mutations:
Theorem 2. Any pair of block diagonal matching field polytopes can be obtained from one another by a sequence of combinatorial mutations such that all intermediate polytopes are matching field polytopes.
The matching fields associated to the intermediate polytopes can be thought of as interpolating between the block diagonal matching fields. As a result, the authors obtain a large family of toric degenerations for the Grassmannian given by matching fields.
Reviewer: Keyvan Yaghmayi (Salt Lake City)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14T99 | Tropical geometry |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
References:
| [1] | Akhtar, M.; Coates, T.; Galkin, S.; Kasprzyk, AM, Minkowski polynomials and mutations, SIGMA. Symm. Integr. Geometry: Methods Appl., 8, 094 (2012) · Zbl 1280.52014 |
| [2] | Alexandersson, P., Gelfand-tsetlin polytopes and the integer decomposition property, Eur. J. Combinat., 54, 1-20 (2016) · Zbl 1331.05028 · doi:10.1016/j.ejc.2015.11.006 |
| [3] | An, BH; Cho, Y.; Kim, JS, On the f-vectors of Gelfand-Tsetlin polytopes, Eur. J. Combinat., 67, 61-77 (2018) · Zbl 1378.52010 · doi:10.1016/j.ejc.2017.07.005 |
| [4] | Anderson, D., Okounkov bodies and toric degenerations, Math. Annalen, 356, 3, 1183-1202 (2013) · Zbl 1273.14104 · doi:10.1007/s00208-012-0880-3 |
| [5] | Bigatti, AM; La Scala, R.; Robbiano, L., Computing toric ideals, J. Symbol. Comput., 27, 4, 351-365 (1999) · Zbl 0958.13009 · doi:10.1006/jsco.1998.0256 |
| [6] | Bossinger, L.; Fang, X.; Fourier, G.; Hering, M.; Lanini, M., Toric degenerations of Gr \((2, n)\) and Gr \((3,6)\) via plabic graphs, Ann. Combinat., 22, 3, 491-512 (2018) · Zbl 1454.14119 · doi:10.1007/s00026-018-0395-z |
| [7] | Bossinger, L.; Lamboglia, S.; Mincheva, K.; Mohammadi, F.; Gregory, G., Computing toric degenerations of flag varieties, Combinatorial Algebraic Geometry, 247-281 (2017), Springer · Zbl 1390.14194 · doi:10.1007/978-1-4939-7486-3_12 |
| [8] | Bossinger, L., Mohammadi, F., Chávez, A.N.: Families of Gröbner degenerations, Grassmannians and universal cluster algebras. arXiv preprintarXiv:2007.14972, (2020) · Zbl 1520.13035 |
| [9] | Chary Bonala, N., Clarke, O., Mohammadi, F.: Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties. arXiv preprint arXiv:2009.03210, (2020) · Zbl 1481.14076 |
| [10] | Chen, T.; Mehta, D., Parallel degree computation for binomial systems, J. Symbol. Comput., 79, 535-558 (2017) · Zbl 1357.65063 · doi:10.1016/j.jsc.2016.07.018 |
| [11] | Clarke, O.; Mohammadi, F., Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux, J. Algebra, 559, 646-678 (2020) · Zbl 1453.14119 · doi:10.1016/j.jalgebra.2020.05.017 |
| [12] | Clarke, O.; Mohammadi, F., Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux, J. Symbol. Comput., 104, 683-723 (2021) · Zbl 1467.14116 · doi:10.1016/j.jsc.2020.09.006 |
| [13] | Clarke, O.; Mohammadi, F., Toric degenerations of flag varieties from matching field tableaux, J. Pure Appl. Algebra, 225, 8, 106624 (2021) · Zbl 1460.14108 · doi:10.1016/j.jpaa.2020.106624 |
| [14] | Cox, DA; Little, JB; Schenck, HK, Toric Varieties (2011), American Mathematical Society · Zbl 1223.14001 · doi:10.1090/gsm/124 |
| [15] | Escobar, L., Harada, M.: Wall-crossing for Newton-Okounkov bodies and the tropical Grassmannian. Int. Math. Res. Notices, rnaa230, (2020) |
| [16] | Fink, A.; Rincón, F., Stiefel tropical linear spaces, J. Combinat. Theory, Ser A, 135, 291-331 (2015) · Zbl 1321.15044 · doi:10.1016/j.jcta.2015.06.001 |
| [17] | Fujita, N., Higashitani, A.: Newton-okounkov bodies of flag varieties and combinatorial mutations. International Mathematics Research Notices, preprintarXiv:2003.10837, (2020) |
| [18] | Galkin, S., Usnich, A.V.: Mutations of potentials. Preprint IPMU 10-0100, (2010) |
| [19] | Gawrilow, E., Joswig, M.: polymake: a framework for analyzing convex polytopes. In: Polytopes—combinatorics and computation (Oberwolfach, 1997), vol. 29 of DMV Sem., P. 43-73. Birkhäuser, Basel, (2000) · Zbl 0960.68182 |
| [20] | Herrmann, S.; Jensen, A.; Joswig, M.; Sturmfels, B., How to draw tropical planes, Electron. J. Combinat., 16, 2, R6 (2009) · Zbl 1195.14080 · doi:10.37236/72 |
| [21] | Higashitani, A.: Two poset polytopes are mutation-equivalent. arXiv:2002.01364, (2020) |
| [22] | Kahle, T., Decompositions of binomial ideals, Ann. Institu. Statist. Math., 62, 4, 727-745 (2010) · Zbl 1440.62394 · doi:10.1007/s10463-010-0290-9 |
| [23] | Kaveh, K.; Manon, C., Khovanskii bases, higher rank valuations, and tropical geometry, SIAM J. Appl. Algebra Geom., 3, 2, 292-336 (2019) · Zbl 1423.13145 · doi:10.1137/17M1160148 |
| [24] | Kogan, M.; Miller, E., Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes, Adv. Math., 193, 1, 1-17 (2005) · Zbl 1084.14049 · doi:10.1016/j.aim.2004.03.017 |
| [25] | Loho, G.; Smith, B., Matching fields and lattice points of simplices, Adv. Math., 370, 107232 (2020) · Zbl 1442.05256 · doi:10.1016/j.aim.2020.107232 |
| [26] | Mohammadi, F.; Shaw, K., Toric degenerations of Grassmannians from matching fields, Algebraic Combinat., 2, 6, 1109-1124 (2019) · Zbl 1504.14089 · doi:10.5802/alco.77 |
| [27] | Rietsch, K.; Williams, L., Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J., 168, 18, 3437-3527 (2019) · Zbl 1439.14142 · doi:10.1215/00127094-2019-0028 |
| [28] | Robbiano, L., Sweedler, M.: Subalgebra bases. In: Bruns, W., Simis, A. (eds.) Commutative Algebra, pp. 61-87. Springer (1990) |
| [29] | Speyer, D.; Sturmfels, B., The tropical Grassmannian, Adv. Geom., 4, 3, 389-411 (2004) · Zbl 1065.14071 · doi:10.1515/advg.2004.023 |
| [30] | Sturmfels, B., Gröbner Bases and Convex Polytopes (1996), American Mathematical Society · Zbl 0856.13020 |
| [31] | Sturmfels, B.; Zelevinsky, A., Maximal minors and their leading terms, Adv. Math., 98, 1, 65-112 (1993) · Zbl 0776.13009 · doi:10.1006/aima.1993.1013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
