Let In,k (respectively, Jn,k) be the number of involutions (respectively, fixed-point free involutions) of {1, . . . , n} with k descents. Motivated by Brenti’s conjecture which states that the sequence In,0, In,1, . . . , In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such t...

The automorphism group AutfSy of the full 2-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup %Z of Aut(Z2) ; %Z...

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...

We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group. Also we present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of...

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 $N$.

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.