Related papers: Viennot shadows and graded module structure in col…

We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and…

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains…

For an ideal $I_{m,n}$ generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this…

P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\surjects B$ by taking a suitable…

Let $R=\bigoplus_{i\geq 0}R_i$ be a Noetherian commutative non-negatively graded ring such that $(R_0,\mathfrak{m}_0)$ is a Henselian local ring. Let $\mathfrak{m}$ be its unique graded maximal ideal $\mathfrak{m}_0+\bigoplus_{i>0}R_i$. Let…

We axiomatize and study the matrices of type $H\in M_N(A)$, having unitary entries, $H_{ij}\in U(A)$, and whose rows and columns are subject to orthogonality type conditions. Here $A$ can be any $C^*$-algebra, for instance $A=\mathbb C$,…

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the…

In this paper we continue our study of modules satisfying the prime radical condition ($\mathbb{P}$-radical modules), that was introduced in Part I (see \cite{BS}). Let $R$ be a commutative ring with identity. The purpose of this paper is…

We provide a real algebraic symbolic-numeric algorithm for computing the real variety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may not be. Our approach uses sets of linear functionals on $R[X]$, vanishing on a given…

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with…

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$ and let $n$ be a non-negative integer. In this paper, we study $\mathcal{S}_{n}(\mathfrak{a})$, a certain class of $A$-modules and we find some sufficient…

Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $\lambda$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by…

We study prime ideals, prime modules, and associated primes of graded modules over rings $S$ graded by a unique product monoid. We consider two situations in detail: (a) the case where $S$ is strongly group-graded and (b) the case where $S$…

Given $n \in \mathbf{N},$ consider the imprimitive wreath product $C_2 \wr S_n.$ We study the structure of modules whose ordinary characters form an involution model of $FC_2 \wr S_n,$ where $F$ is a field of odd prime characteristic. We…

Centrosymmetric matrices of order $n$ over an arbitrary algebra $R$ form a subalgebra of the full $n\times n$ matrix algebra over $R$. It is called the centrosymmetric matrix algebra of order $n$ over $R$ and denoted by $S_n(R)$. We prove…

We study the restriction to the symmetric group, $\mc{S}_n$ of the adjoint representation of $\mt{GL}_n(\C)$. We determine the irreducible constituents of the space of symmetric as well as the space of skew-symmetric $n\times n$ matrices as…

This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next…

The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular…

We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring, and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a…

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak…