Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega \left(𝒟\right)$ be the Burnside ring of $𝒟$, $\Delta \left(𝒟\right)$ be the augmentation ideal of $\Omega \left(𝒟\right)$. Denote by ${\Delta }^n\left(𝒟\right)$ and $Q_n\left(𝒟\right)$ the $n$th power of $\Delta \left(𝒟\right)$ and the $n$th consecutive quotient group ${\Delta }^n\left(𝒟\right)/{\Delta }^{n+1}\left(𝒟\right)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for ${\Delta }^n\left(𝒟\right)$ and determines the isomorphism class of $Q_n\left(𝒟\right)$ for each positive integer $n$.

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation $\rho$ of the Galois group of a number field $F$ as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over $F$, provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification...

Suppose $R$ is a commutative ring with identity of prime characteristic $p$ and $G$ is an arbitrary abelian $p$-group. In the present paper, a basic subgroup and a lower basic subgroup of the $p$-component $U_p\left(RG\right)$ and of the factor-group $U_p\left(RG\right)/G$ of the unit group $U\left(RG\right)$ in the modular group algebra $RG$ are established, in the case when $R$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $p$-component $S\left(RG\right)$ and of the quotient group $S\left(RG\right)/G_p$ are given when $R$ is perfect and $G$ is arbitrary whose $G/G_p$ is $p$-divisible....

Let $S\left(RG\right)$ be a normed Sylow $p$-subgroup in a group ring $RG$ of an abelian group $G$ with $p$-component $G_p$ and a $p$-basic subgroup $B$ over a commutative unitary ring $R$ with prime characteristic $p$. The first central result is that $1+I(RG;B_p)+I(R\left(p^i\right)G;G)$ is basic in $S\left(RG\right)$ and $B[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]$ is $p$-basic in $V\left(RG\right)$, and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G_p/G_p$ is basic in $S\left(RG\right)/G_p$ and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G/G$ is $p$-basic in $V\left(RG\right)/G$, provided in both cases $G/G_p$ is $p$-divisible and $R$ is such that its maximal perfect subring $R^{p^i}$ has no nilpotents whenever $i$ is natural. The second major result is that $B(1+I(RG;B_p))$ is $p$-basic in $V\left(RG\right)$ and $(1+I(RG;B_p))G/G$ is $p$-basic in $V\left(RG\right)/G$,...

Suppose $F$ is a perfect field of $\mathrm{c}harF=p\ne 0$ and $G$ is an arbitrary abelian multiplicative group with a $p$-basic subgroup $B$ and $p$-component $G_p$. Let $FG$ be the group algebra with normed group of all units $V\left(FG\right)$ and its Sylow $p$-subgroup $S\left(FG\right)$, and let $I_p(FG;B)$ be the nilradical of the relative augmentation ideal $I(FG;B)$ of $FG$ with respect to $B$. The main results that motivate this article are that $1+I_p(FG;B)$ is basic in $S\left(FG\right)$, and $B(1+I_p(FG;B))$ is $p$-basic in $V\left(FG\right)$ provided $G$ is $p$-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...

We demonstrate the way in which composition of two famous combinatorial bijections, of Robinson-Schensted and Kerov-Kirillov-Reshetikhin, applied to the Heisenberg model of magnetic ring with spin 1/2, defines the geography of rigged strings (which label exact eigenfunctions of the Bethe Ansatz) on the classical configuration space (the set of all positions of the system of r reversed spins). We point out that each l-string originates, in the language of this bijection, from an island of l consecutive...

The set of invariant symmetric bilinear forms on irreducible modules over fields of characteristic zero for certain groups is studied. Results are obtained under the presence in a finite group of elements of order four whose square is central. In particular, we find that the relevant modules for the groups mentioned in the title always accept an invariant symmetric bilinear form under which the module admits an orthonormal basis.

The present paper introduces a group of transformations on the collection of all bivariate copulas. This group contains an involution which is particularly useful since it provides (1) a criterion under which a given symmetric copula can be transformed into an asymmetric one and (2) a condition under which for a given copula the value of every measure of concordance is equal to zero. The group also contains a subgroup which is of particular interest since its four elements preserve symmetry, the...