Mathematics – Commutative Algebra
Scientific paper
2010-02-23
Discrete Math. 310 (2010), no. 24, 3558-3570
Mathematics
Commutative Algebra
16 pages. Minor revisions, mainly to keep track of two interesting developments following the original posting. Final version
Scientific paper
We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in characteristic p. We wonder how powerful can be this connection between combinatorial commutative algebra and partition theory. We present a first result in this direction, by deducing, using our algebraic techniques for the WLP, some explicit information on the rational primes dividing M(a,b,c).
Li Jizhou
Zanello Fabrizio
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