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September 30, 2025Open Access

The F-singularities of algebras defined by permanents

Key Points

The study reveals that $Bbbk[X]/P_n(X)$ is $F$-regular when $ ext{char} Bbbk > 2$, similar to the case where $ ext{char} Bbbk = 2$.
A full characterization of $F$-purity and $F$-regularity for $Bbbk[X]/P_2(X)$ is provided when $ ext{char} Bbbk > 2$, differing from the even characteristic case.
The paper addresses algebras derived from symmetric or Hankel matrices of indeterminates and their properties based on their respective characteristics.
It highlights the significance of determinantal ideals, especially in commutative algebra and algebraic geometry contexts.

Abstract

Let X be a matrix of indeterminates, t an integer, and Pₜ (X) define the ideal generated by the permanents of all t t submatrix of X. Pₜ (X) is called a permanental ideal. In this article, we study the algebras /Pₜ (X) where X is a generic, symmetric, or a Hankel matrix of indeterminates. When char = 2, Pₜ (X) is also known as a determinantal ideal, a popular class in commutative algebra and algebraic geometry, and thus many properties of Pₜ (X) are known in this case. We prove that, if X is an n n matrix and char >2, the algebra /Pₙ (X) is F-regular, just like when char = 2. On the other hand, we obtain a full characterization of when /P₂ (X) is F-pure or F-regular, when char >2, and the answer is different than that in even characteristic.

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Cite This Study

Trung Chau (Fri,) studied this question.

synapsesocial.com/papers/68dc1e308a7d58c25ebb13f7 https://doi.org/https://doi.org/10.48550/arxiv.2509.09980

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