Related papers: The ring of perfect $p$-permutation bimodules for …

Using the properties of the ideal of the coordinate Hermite interpolation on n-dimensional grid [4], we prove that the extension k in k[x1, x2, ..., xn] / (f1(x1), ..., fn(xn)) has a primitive element if and only if at most one of the…

Let $p$ be an odd prime and let $k$ be a field of characteristic $p$. We provide a practical algebraic description of the representation ring of $k\mathrm{SL}_2(\mathbb{F}_p)$ modulo projectives. We then investigate a family of modular…

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$,…

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parametrized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a…

We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a…

We provide a series of examples of finite groups G and mod p representations V of G whose stable endomorphisms are all given by scalars such that V has a universal deformation ring R(G,V) which is large in the sense that R(G,V)/pR(G,V) is…

In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},\ldots , a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{\sigma (1)} a_{\sigma (2)} \cdots a_{\sigma (n)}$, where…

We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian…

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal…

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…

For $G$ a finite group, $k$ a field of prime characteristic $p$, and $S$ a Sylow $p$-subgroup of $G$, the Sylow permutation module $\mathrm{Ind}^G_S(k)$ plays a role in diverse facets of representation theory and group theory, ranging from…

In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and…

For a fixed positive integer $k$, any element $g$ of the permutation group $S_{k}$ acts on the tensor product vertex operator algebra $V^{\otimes k}$ in the obvious way. In this paper, we determine the $S$-matrix of $\left(V^{\otimes…

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…

Let $R$ be a local ring of characteristic $p>0$ which is $F$-finite and has perfect residue field. We compute the generalized Hilbert-Kunz invariant for certain modules over several classes of rings: hypersurfaces of finite representation…

For an arbitrary ring $A$, we study the abelianization of the elementary group $\textrm{E}_2(A)$. In particular, we show that for a commutative ring $A$ there exists an exact sequence \[ K_2(2,A)/C(2,A) \to A/M \to…

Let R be a unitary ring of finite cardinality P^k, where p is a prime number and $p\nmid k$. We show that if the group of units of $R$ has at most one subgroup of order $p$, then $R\cong A\bigoplus B,$ where $B$ is a finite ring of order…

Integral cluster categories of acyclic quivers have recently been used in the representation-theoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would…

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…