Illinois Journal of Mathematics

Ideals generated by principal minors

Ashley K. Wheeler

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A minor is principal means it is defined by the same row and column indices. We study ideals generated by principal minors of size $t\leq n$ of a generic $n\times n$ matrix $X$, in the polynomial ring generated over an algebraically closed field by the entries of $X$. When $t=2$ the resulting quotient ring is a normal complete intersection domain. We show for any $t$, upon inverting $\det X$ the ideals given respectively by the size $t$ and the size $n-t$ principal minors become isomorphic. From that we show the algebraic set given by the size $n-1$ principal minors has a codimension $4$ component defined by the determinantal ideal, plus a codimension $n$ component. When $n=4$ the two components are linked, and in fact, geometrically linked.

Article information

Illinois J. Math., Volume 59, Number 3 (2015), 675-689.

Received: 1 January 2015
Revised: 28 March 2016
First available in Project Euclid: 30 September 2016

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Zentralblatt MATH identifier

Primary: 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 14M12: Determinantal varieties [See also 13C40]


Wheeler, Ashley K. Ideals generated by principal minors. Illinois J. Math. 59 (2015), no. 3, 675--689. doi:10.1215/ijm/1475266403.

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