Related papers: When is a Specht ideal Cohen-Macaulay?

Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$…

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show…

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…

This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I \subset R. It is known that there are embeddings…

For $\Bbbk$ a field, let $X$ a $m \times n$ matrix of variables and $S=\Bbbk[X].$ We consider the determinantal ideal $I_2 \subseteq S$ generated by the $2$-minors of $X.$ In this paper we find a suitable monomial order over $S$ such that…

The third named author and P\'{e}rez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the…

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p >…

Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda…

Let kappa a regular uncountable cardinal and lambda a cardinal >kappa, and suppose lambda^{<kappa} is less than the covering number for category cov(M_{kappa,kappa}). Then (a) I_{kappa,lambda}^+ -->^kappa (I_{kappa, lambda}^+,omega +1)^2,…

The permutation matrices form a subgroup of $\text{GL}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{\mu\lambda}$ denote the multiplicity of the irreducible representation $V_\mu$ of $S_n$, corresponding to a…

Given $\Sigma\subset\mathbb K[x_1,\ldots,x_k]$, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, it has been conjectured that $I_a(\Sigma)$, the ideal generated by all $a$-fold products of…

We consider a class of graphs $G$ such that the height of the edge ideal $I(G)$ is half of the number $\sharp V(G)$ of the vertices. We give Cohen-Macaulay criteria for such graphs.

In this paper it is proved that the ideal $I_w$ of the weak polynomial identities of the superalgebra $M_{1,1}(E)$ is generated by the proper polynomials $[x_1,x_2,x_3]$ and $[x_2,x_1][x_3,x_1][x_4,x_1]$. This is proved for any infinite…

We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $\kappa \leq \lambda$, where $\kappa$ is the successor of a regular cardinal, there is a rigid saturated ideal on…

Over a field of characteristic 0, every ring of invariants of a finite group is Cohen-Macaulay. This is not true for fields of positive characteristic. We consider permutation representations and their invariant rings over fields…

Let $I_\mathbb{X}$ be the bihomogeneous ideal of a finite set of points $\mathbb{X} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$. The purpose of this note is to consider ``splittings'' of the ideal $I_\mathbb{X}$, that is, finding ideals $J$…

Given an integer $n\geq 1$ and an irreducible character $\chi_{\lambda}$ of $S_{n}$ for some partition $\lambda$ of $n$, the immanant $\mathrm{imm}_{\lambda}:\mathbb{C}^{n\times n}\to\mathbb{C}$ maps matrices $A\in\mathbb{C}^{n\times n}$ to…

We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} \alpha^{-1} (\lambda_1 +\cdots +\lambda_{\alpha}) > - \theta \bar{\lambda}, \end{equation*}…

We prove that for any partition $(\lambda_1,...,\lambda_{d^2})$ of size $\ell d$ there exists $k\ge 1$ such that the tensor square of the irreducible representation of the symmetric group $S_{k\ell d}$ with respect to the rectangular…

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1, \ldots, x_n]$, where $k$ is a field such that Char$(k)$ does not divide $n!$. We use Higher Specht polynomials to construct matrix factorizations…