Wavefunctions of symmetrical nanoparticles are considered making use of induced representation method. It is shown that when, at the same total symmetry, the order of local symmetry group decreases, additional quantum numbers are required for complete labelling of electron states. It is shown that the labels of irreducible representations of intermediate subgroups can be used for complete classification of states in the case of repeating IRs in symmetry adapted linear combinations. The intermediate...

A fairly old problem in modular representation theory is to determine the vanishing behavior of the $\mathrm{H}om$ groups and higher $\mathrm{E}xt$ groups of Weyl modules and to compute the dimension of the $\mathbb{Z}/\left(p\right)$-vector space ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ for any partitions $\lambda$, $\mu$ of $r$, which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ and provide a new formula for the intertwining number for any...

Let $𝔽$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $𝔽$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,...,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $S{L}_2\left(ℂ\right)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

Nous étudions le problème de l’irréductibilité du produit tensoriel de deux représentations irréductibles d’un groupe fondamental $G={\pi }_1\left(X\right)$, quand $X$ est le complémentaire d’hypersurfaces dans un espace projectif. Nous mettons en place un formalisme adapté et utilisons une approche par monodromie pour définir une classe de représentations irréductibles de $G$ dont les produits tensoriels restent irréductibles pour des valeurs génériques de paramètres de définition. Ceci est appliqué au groupe de tresses pures...

Suppose $G$ is a $p$-mixed splitting abelian group and $R$ is a commutative unitary ring of zero characteristic such that the prime number $p$ satisfies $p\notin \text{inv}\left(R\right)\cup \text{zd}\left(R\right)$. Then $R\left(H\right)$ and $R\left(G\right)$ are canonically isomorphic $R$-group algebras for any group $H$ precisely when $H$ and $G$ are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995).

∗ The work was supported by the National Fund “Scientific researches” and by the Ministry of Education and Science in Bulgaria under contract MM 70/91.Let K be a field of characteristic p > 0 and let G be a direct sum of cyclic groups, such that its torsion part is a p-group. If there exists a K-isomorphism KH ∼= KG for some group H, then it is shown that H ∼= G. Let G be a direct sum of cyclic groups, a divisible group or a simply presented torsion abelian group. Then KH ∼= KG as K-algebras...