相关论文: Differential symmetric signature in high dimension
The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the…
We define two related invariants for a $d$-dimensional local ring $(R,\mathfrak{m},k)$ called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top…
This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of…
The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to…
We define the dual F-signature of modules, which is equivalent to the F-signature if the module is the base ring. By using this invariant, We give characterizations of regular, F-regular, F-rational, and Gorenstein singularities.
We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the…
The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free…
It is known that a certain invariant subring $R$ has finite $F$-representation type. Thus, we can write the $R$-module ${}^eR$ as a finite direct sum of finitely many $R$-modules. In such a decomposition of ${}^eR$, we pay attention to the…
For a commutative ring $R$, the $F$-signature was defined by Huneke and Leuschke \cite{H-L}. It is an invariant that measures the order of the rank of the free direct summand of $R^{(e)}$. Here, $R^{(e)}$ is $R$ itself, regarded as an…
Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild…
In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{\Gamma}$ where $\Gamma$ is a product of general linear groups over a field…
Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings…
The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules and called dual $F$-signature. In this paper, we…
The $F$-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong $F$-regularity, an important class of singularities related to KLT singularities in characteristic zero. In…
Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
We compute the diagonal F-thresholds of determinantal hypersurfaces arising from a generic matrix and from a generic symmetric matrix, as well as of the Pfaffian hypersurface arising from a generic skew-symmetric matrix of even size. The…
We present a closed formula, analogous to the Weyl dimension formula, for the signature of an invariant Hermitian form on any finite-dimensional irreducible representation of a real reductive Lie group, assuming that such a form exists. The…
We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of $SU(2)$. We introduce the asymptotic positivity ratio and…
Let n>0 be an integer and let B_{n} denote the hyperoctahedral group of rank n. The group B_{n} acts on the polynomial ring Q[x_{1},...,x_{n},y_{1},...,y_{n}] by signed permutations simultaneously on both of the sets of variables…