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1. CJM 2016 (vol 69 pp. 687)

Ovchinnikov, Alexey; Wibmer, Michael
Tannakian Categories with Semigroup Actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.

Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06

2. CJM 2016 (vol 68 pp. 541)

Garcia-Armas, Mario
Strongly Incompressible Curves
Let $G$ be a finite group. A faithful $G$-variety $X$ is called strongly incompressible if every dominant $G$-equivariant rational map of $X$ onto another faithful $G$-variety $Y$ is birational. We settle the problem of existence of strongly incompressible $G$-curves for any finite group $G$ and any base field $k$ of characteristic zero.

Keywords:algebraic curves, group actions, Galois cohomology
Categories:14L30, 14E07, 14H37

3. CJM 2015 (vol 67 pp. 1247)

Barros, Carlos Braga; Rocha, Victor; Souza, Josiney
Lyapunov Stability and Attraction Under Equivariant Maps
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume that $\mathcal{S}$ is a semigroup acting on both $M$ and $N$. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors and Lyapunov stable sets (all concepts defined for the action of the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations from $M$ to $N$.

Keywords:Lyapunov stability, semigroup actions, generalized flows, equivariant maps, admissible topological spaces
Categories:37B25, 37C75, 34C27, 34D05

4. CJM 2003 (vol 55 pp. 693)

Borne, Niels
Une formule de Riemann-Roch équivariante pour les courbes
Soit $G$ un groupe fini agissant sur une courbe alg\'ebrique projective et lisse $X$ sur un corps alg\'ebriquement clos $k$. Dans cet article, on donne une formule de Riemann-Roch pour la caract\'eristique d'Euler \'equivariante d'un $G$-faisceau inversible $\mathcal{L}$, \`a valeurs dans l'anneau $R_k (G)$ des caract\`eres du groupe $G$. La formule donn\'ee a un bon comportement fonctoriel en ce sens qu'elle rel\`eve la formule classique le long du morphisme $\dim \colon R_k (G) \to \mathbb{Z}$, et est valable m\^eme pour une action sauvage. En guise d'application, on montre comment calculer explicitement le caract\`ere de l'espace des sections globales d'une large classe de $G$-faisceaux inversibles, en s'attardant sur le cas particulier d\'elicat du faisceau des diff\'erentielles sur la courbe.

Keywords:group actions on varieties or schemes,, Riemann-Roch theorems
Categories:14L30, 14C40

5. CJM 2002 (vol 54 pp. 1254)

Isaev, A. V.; Kruzhilin, N. G.
Effective Actions of the Unitary Group on Complex Manifolds
We classify all connected $n$-dimensional complex manifolds admitting effective actions of the unitary group $U_n$ by biholomorphic transformations. One consequence of this classification is a characterization of $\CC^n$ by its automorphism group.

Keywords:complex manifolds, group actions
Categories:32Q57, 32M17

6. CJM 2002 (vol 54 pp. 571)

Li, Chi-Kwong; Poon, Yiu-Tung
Diagonals and Partial Diagonals of Sum of Matrices
Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a certain group action such as \begin{enumerate}[(4)] \item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices $A$ by $(U,V)*A = UAV^t$, \item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m \times n$ real matrices $A$ by $(U,V)*A = UAV^t$, \item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$, \item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$. \end{enumerate} Denote by $$ \mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in \mathcal{O}(A_i), i = 1,\dots,k\} $$ the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$, {\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1, \dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of $A_1,\dots,A_k$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex Hermitian matrices under the action of unitary similarities are also discussed.

Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular values
Categories:15A42, 15A18

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