The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a $\lambda$-ideal of the special $\lambda$-ring of symmetric group class functions.

Given a tuple $({𝒳}_1,...,{𝒳}_k)$ of irreducible characters of $G{L}_n\left(F_q\right)$ we define a star-shaped quiver $\Gamma$ together with a dimension vector $v$. Assume that $({𝒳}_1,...,{𝒳}_k)$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product ${𝒳}_1\otimes \cdots \otimes {𝒳}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we...

On présente deux résultats nouveaux concernant la racine carrée de la codifférente d’une extension faiblement ramifiée de $\mathbb{Q}$. Le premier, relatif à sa structure galoisienne, s’inscrit dans la stratégie classique développée notamment par Fröhlich et Taylor. Le second, qui concerne le réseau entier unimodulaire associé, est prouvé à l’aide de calculs numériques portant sur des exemples intéressants.

On introduit une opérade anticyclique $\mathbf{V}$ définie par une présentation ternaire quadratique. On montre qu’elle admet une base indexée par les arbres binaires planaires. On relie cette construction à la famille des treillis de Tamari ${\left({\mathsf{Y}}_n\right)}_{n\ge 0}$ en construisant un isomorphisme entre $\mathbf{V}(2n+1)$ et le groupe de Grothendieck de la catégorie $mod{\mathsf{Y}}_n$ qui envoie la base de $\mathbf{V}(2n+1)$ sur les classes des modules projectifs et qui transforme la structure anticyclique de $\mathbf{V}$ en la transformation de Coxeter de la catégorie dérivée de $mod{\mathsf{Y}}_n$. La dualité...

Let $p$ be a prime number. This paper introduces the Roquette category ${ℛ}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In ${ℛ}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called the edge of $P$. Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$-groups, and the tensor structure of ${ℛ}_p$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors from ${ℛ}_p$ to...

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW-complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses invariants like Lefschetz number, its generalization to the Lefschetz invariant, Nielsen number and $L^2$-torsion of mapping tori. We examine its behaviour under fibrations.

We prove that for any ring $𝐑$ of Krull dimension not greater than 1 and $n\ge 3$, the group ${\mathrm{E}}_n\left(𝐑[X,X^{-1}]\right)$ acts transitively on ${\mathrm{Um}}_n\left(𝐑[X,X^{-1}]\right)$. In particular, we obtain that for any ring $𝐑$ with Krull dimension not greater than 1, all finitely generated stably free modules over $𝐑[X,X^{-1}]$ are free. All the obtained results are proved constructively.