A fixed-point free involution on a finite set S is defined as a bijection I : S → S such as ∀e ∈ S, I(I(e)) = e and ∀e ∈ S, I(e) 6= e. In this article, the fixed-point free involutions are represented as partitions of the set S, and some properties linked to this representation are exhibited. Then an optimal algorithm to list all the fixed-point free involutions is presented. Its soundess relie...

Several authors have examined connections among restricted permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for involutions which avoid 3412. Our results include a recursive procedure for computing the generating function for involutions which avoid 3412 and any set of additional patterns. We use our results to gi...

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with invo...

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the qui...

This paper establishes an equivalence between existence of free involutions on HCP 3 and existence of involutions on S with fixed point set an imbedded S, then a family of counterexamples of the Smith conjecture for imbeddings of S in S are given by known result on HCP . In addition, this paper also shows that every smooth homotopy complex projective 3-space admits no orientation preserving smo...

This paper establishes an equivalence between existence of free involutions on HCP 3 and existence of involutions on S with fixed point set an imbedded S, then a family of counterexamples of the Smith conjecture for imbeddings of S in S are given by known result on HCP . In addition, this paper also shows that every smooth homotopy complex projective 3-space admits no orientation preserving smo...