Permanental ideals of symmetric matrices

In this article, we study the ideal generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 times 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> × </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2\times 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> permanents of a symmetric matrix. We denote this ideal by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P 2 left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P_2(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a symmetric matrix. We compute a Gröbner basis, dimension, depth, minimal primes, and a primary decomposition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P 2 left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P_2(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.