Permanental ideals (Q1582302)

The permanent of an \((n\times n)\) square matrix \(M=(a_{ij})\) is defined as \[ \text{perm}(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)} a_{2\sigma(2)}\cdots a_{n\sigma(n)}. \] The permanent differs from the determinant only in the lack of minus signs in the expansion. In this paper permanental ideals of generic matrices are studied. Let \(P_{r}(M)\) denote the permanental ideal generated by all the \((r \times r)\)-subpermanents of a generic \((m\times n)\) matrix \(M\) of indeterminates over a field \(F\). If the characteristic of \(F\) is \(2\), then permanental ideals coincide with determinantal ideals, which are well-understood [see \textit{W. Bruns} and \textit{U. Vetter}, ``Determinantal rings'', Lect. Notes Math. 1327 (1988; Zbl 0673.13006)]. In the paper under review the case \(r=2\) is considered, when the characteristic of \(F\) is different from \(2\). It turns out that, in this case, permanental ideals behave very differently from determinantal ideals. For example, it is shown that if \(m,n\geq 3\), then \(P_{2}(M)\) is not a radical ideal, is not Cohen-Macaulay, and there are minimal primes of distinct heights over it. Moreover if \(m,n\geq 4\), then \(P_2(M)\) is not integrally closed. Primary decomposition, the unmixed parts of all possible dimensions, and the radical of \(P_{2}(M)\) are explicitly calculated. Furthermore, a Gröbner basis for \(P_{2}(M)\) with respect to any diagonal order is given. Unlike in the determinantal case, such a Gröbner basis consists not only of the permanents. Thus, the permanental ideals are a case of ideals for which Gröbner bases, irreducibility, primary decompositions, Cohen-Macaulayness, and integral-closedness depend on the characteristic of the underlying field: They have good properties in characteristic \(2\) versus very different properties in all other characteristics.