Permanental ideals of symmetric matrices

In this article, we study the ideal generated by 2 × 2 2× 2 permanents of a symmetric matrix. We denote this ideal by P 2 ( X ) P_2(X) where X X is a symmetric matrix. We compute a Gröbner basis, dimension, depth, minimal primes, and a primary decomposition of P 2 ( X ) P_2(X) . It can be seen that the answer is reliant on whether the characteristic of the base field is two, and thus these ideals constitute a class of ideals whose algebraic properties depend on characteristics of the base field.