ALGEBRAIC COMBINATORICS

no. 1

Lower bound cluster algebras: presentations, Cohen–Macaulayness, and normality
Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 95-114.

We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Gröbner degenerates to the Stanley–Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen–Macaulay. Finally, we use Stanley–Reisner combinatorics and a result of Knutson–Lam–Speyer to show that all lower bound algebras are normal.

Received: 2017-08-02
Accepted: 2017-08-18
Published online: 2018-01-29
DOI: https://doi.org/10.5802/alco.2
Classification: 13F60,  05E40,  13F55
Keywords: Cluster algebras, lower bound cluster algebras, combinatorial commutative algebra, Stanley–Reisner complexes 
@article{ALCO_2018__1_1_95_0,
     author = {Muller, Greg and Rajchgot, Jenna and Zykoski, Bradley},
     title = {Lower bound cluster algebras: presentations, Cohen--Macaulayness, and normality},
     journal = {Algebraic Combinatorics},
     pages = {95--114},
     publisher = {MathOA foundation},
     volume = {1},
     number = {1},
     year = {2018},
     doi = {10.5802/alco.2},
     zbl = {06882336},
     mrnumber = {3857161},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2018__1_1_95_0/}
}
Muller, Greg; Rajchgot, Jenna; Zykoski, Bradley. Lower bound cluster algebras: presentations, Cohen–Macaulayness, and normality. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 95-114. doi : 10.5802/alco.2. https://alco.centre-mersenne.org/item/ALCO_2018__1_1_95_0/

[1] Benito, Angélica; Muller, Greg; Rajchgot, Jenna; Smith, Karen E. Singularities of locally acyclic cluster algebras, Algebra Number Theory, Volume 9 (2015) no. 4, pp. 913-936 | Article | MR 3352824 | Zbl 1350.13017

[2] Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., Volume 126 (2005) no. 1, pp. 1-52 | Article | MR 2110627 | Zbl 1135.16013

[3] Billera, Louis J.; Provan, J. Scott A decomposition property for simplicial complexes and its relation to diameters and shellings, Second International Conference on Combinatorial Mathematics (New York, 1978) (Ann. New York Acad. Sci.) Volume 319, New York Acad. Sci., New York, 1979, pp. 82-85 | MR 556009 | Zbl 0484.52006

[4] Björner, Anders; Wachs, Michelle L. Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc., Volume 349 (1997) no. 10, pp. 3945-3975 | Article | MR 1401765 | Zbl 0886.05126

[5] Bruns, Winfried; Conca, Aldo Groebner bases, initial ideals and initial algebras, 2003 (arXiv:math/0308102)

[6] Brüstle, Thomas; Dupont, Grégoire; Pérotin, Matthieu On maximal green sequences, Int. Math. Res. Not. (2014) no. 16, pp. 4547-4586 | Article | MR 3250044 | Zbl 1346.16009

[7] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, Volume 150, Springer-Verlag, New York, 1995, xvi+785 pages | MR 1322960 | Zbl 0819.13001

[8] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | Article | MR 1887642 | Zbl 1021.16017

[9] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | Article | MR 2295199 | Zbl 1127.16023

[10] Geiss, Christof; Leclerc, Bernard; Schröer, Jan Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 3, pp. 825-876 | Article | Numdam | MR 2427512 | Zbl 1151.16009

[11] Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek Cluster algebras and Weil-Petersson forms, Duke Math. J., Volume 127 (2005) no. 2, pp. 291-311 | Article | MR 2130414 | Zbl 1079.53124

[12] Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim Canonical bases for cluster algebras, 2014 (arXiv:1411.1394) | Zbl 06836101

[13] Knutson, Allen Frobenius splitting, point-counting, and degeneration, 2009 (arXiv:0911.4941)

[14] Knutson, Allen; Lam, Thomas; Speyer, David E. Projections of Richardson varieties, J. Reine Angew. Math., Volume 687 (2014), pp. 133-157 | Article | MR 3176610 | Zbl 1345.14047

[15] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, Volume 227, Springer-Verlag, New York, 2005, xiv+417 pages | MR 2110098 | Zbl 1066.13001

[16] Muller, Greg Locally acyclic cluster algebras, Adv. Math., Volume 233 (2013), pp. 207-247 | Article | MR 2995670 | Zbl 1279.13032

[17] Muller, Greg A=U for locally acyclic cluster algebras, SIGMA, Volume 10 (2014), 094, 8 pages | Article | MR 3261850 | Zbl 1327.13079

[18] Muller, Greg The existence of a maximal green sequence is not invariant under quiver mutation, 2015 (arXiv:1503.04675) | Zbl 1339.05163

[19] Munkres, James R. Topological results in combinatorics, Michigan Math. J., Volume 31 (1984) no. 1, pp. 113-128 | Article | MR 736476 | Zbl 0585.57014

[20] Scott, J. S. Grassmannians and cluster algebras, Proc. London Math. Soc. (3), Volume 92 (2006) no. 2, pp. 345-380 | Article | MR 2205721 | Zbl 1088.22009