The $F$-singularities of algebras defined by permanents

Abstract

Let $X$ be a matrix of indeterminates, $t$ an integer, and $P_t(X)$ define the ideal generated by the permanents of all $t\times t$ submatrix of $X$. $P_t(X)$ is called a permanental ideal. In this article, we study the algebras $\Bbbk[X]/P_t(X)$ where $X$ is a generic, symmetric, or a Hankel matrix of indeterminates. When $\operatorname{char}\Bbbk = 2$, $P_t(X)$ is also known as a determinantal ideal, a popular class in commutative algebra and algebraic geometry, and thus many properties of $P_t(X)$ are known in this case. We prove that, if $X$ is an $n\times n$ matrix and $\operatorname{char} \Bbbk>2$, the algebra $\Bbbk[X]/P_n(X)$ is $F$-regular, just like when $\operatorname{char} \Bbbk = 2$. On the other hand, we obtain a full characterization of when $\Bbbk[X]/P_2(X)$ is $F$-pure or $F$-regular, when $\operatorname{char} \Bbbk >2$, and the answer is different than that in even characteristic.