Introduction. Let k be a field of characteristic not two and G a connected linear reductive k-group. By a k-involution θ of G, we mean a k-automorphism θ of G of order two. For k = R, C or an algebraically closed field, such involutions have been extensively studied emerging from different interests. As manifested in [8, 18, 28], the interactions with the representation theory of reductive grou...

In this paper, we study a special class of elements in the finite commutative rings called involutions. An involution ring R is an element with property that x^2-1=0 for some x R. This describes both implementation and enumeration involutions various rings, such as cyclic non-cyclic zero-rings, fields, especially Gaussian integers. The paper begins simple well-known results equation over It pro...

There are two types of numerically trivial involutions of an Enriques surface according as their period lattice. One is U(2) ⊥ U(2)-type and the other is U ⊥ U(2)-type. An Enriques surface with an involution of U(2) ⊥ U(2)-type is doubly covered by a Kummer surface of product type, and such involutions are classified again into two types according as the parity of the corresponding Göpel subgro...

A finite subgroup G of GL(n,C) is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements g ∈ G such that gḡ = 1, where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including,...

We determine the automorphism group of modular curve $X_0^*(N)$, obtained as quotient $X_0(N)$ by its Atkin-Lehner involutions, for all square-free values $N$.

Abstract Starting from an anti-symplectic involution on a K3 surface, one can consider natural Lagrangian subvariety inside the moduli space of sheaves over K3. One also construct Prymian integrable system following construction Markushevich–Tikhomirov, extended by Arbarello–Saccà–Ferretti, Matteini and Sawon–Shen. In this article we address question Sawon, showing that these systems their asso...

In this paper we study the conditions under which an involution becomes metabolic over a quadratic field extension. We characterise those involutions that become metabolic over a given separable quadratic extension. We further give an example of an anisotropic orthogonal involution that becomes isotropic over a separable quadratic extension.

Let W be a finite Coxeter group and X a subset of W . The length polynomial LW,X(t) is defined by LW,X(t) = ∑ x∈X t `(x), where ` is the length function on W . In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group W . In particular, these results correct errors in [6] for the invo...

We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a ...