Illinois Journal of Mathematics

A spherical initial ideal for Pfaffians

Jakob Jonsson and Volkmar Welker

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We determine a term order on the monomials in the variables $\varx{i}{j}$, $1 \leq i < j \leq n$, such that corresponding initial ideal of the ideal of Pfaffians of degree $r$ of a generic $n$ by $n$ skew-symmetric matrix is the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. Moreover, we demonstrate that the Pfaffians of the $2r$ by $2r$ skew-symmetric submatrices form a Gröobner basis for the given term order. The same methods and similar term orders as for the Pfaffians also yield squarefree initial ideals for certain determinantal ideals. Yet, in contrast to the case of Pfaffians, the corresponding simplicial complexes are balls that do not decompose into a join as above.

Article information

Illinois J. Math., Volume 51, Number 4 (2007), 1397-1407.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12]
Secondary: 05E99: None of the above, but in this section 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10]


Jonsson, Jakob; Welker, Volkmar. A spherical initial ideal for Pfaffians. Illinois J. Math. 51 (2007), no. 4, 1397--1407. doi:10.1215/ijm/1258138551.

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