## Illinois Journal of Mathematics

### A spherical initial ideal for Pfaffians

#### Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138551

Digital Object Identifier
doi:10.1215/ijm/1258138551

Mathematical Reviews number (MathSciNet)
MR2417434

Zentralblatt MATH identifier
1148.13012

#### Citation

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