On the Stanley depth of squarefree Veronese ideals |
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Authors: | Mitchel T Keller Yi-Huang Shen Noah Streib Stephen J Young |
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Institution: | 1.School of Mathematics,Georgia Institute of Technology,Atlanta,USA;2.Department of Mathematics,University of Science and Technology of China,Hefei,China;3.Department of Mathematics,University of California, San Diego,La Jolla,USA |
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Abstract: | Let K be a field and S=Kx
1,…,x
n
]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in
the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze
squarefree Veronese ideals in S. In particular, if I
n,d
is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I
n,d
)=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I
n,d
)≤⌊(n−d)/(d+1)⌋+d. |
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Keywords: | |
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