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.
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/
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