Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 59, Number 3 (2015), 675-689.
Ideals generated by principal minors
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.
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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Wheeler, Ashley K. Ideals generated by principal minors. Illinois J. Math. 59 (2015), no. 3, 675--689. doi:10.1215/ijm/1475266403. https://projecteuclid.org/euclid.ijm/1475266403