Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Constructing Pseudo-Random Permutations with a Prescribed Structure

We show how to construct pseudo-random permutations that satisfy a certain cycle restriction, for example that the permutation be cyclic (consisting of one cycle containing all the elements) or an involution (a self-inverse permutation) with no fixed points. The construction can be based on any (unrestricted) pseudo-random permutation. The resulting permutations are defined succinctly and their evaluation at a given point is efficient. Furthermore, they enjoy a fast forward property, i.e. it is possible to iterate them at a very small cost. Received 10 August 2000 and revised 30 September 2000 Online publication 9 April 2001     

A unification of some matrix factorization results

A unification of some factorization results regarding products of positive-definite matrices, commutators, and products of involutions is presented. This topic was first presented by Sourour for fields with, in the latter two classes, sufficiently many elements in terms of the order of the matrix being factored. The current presentation is valid for matrices over any field with at least four elements, and is independent of the order of the matrix being factored.     

不动点集为RP（1）×CP（N）的带有对合的流形

令 (M,T)是一个在光滑闭流形上的光滑对合 ,它的不动点集为 F ,本文确定了 F=RP(1 )× CP(N )的对合的协边分类     

Permutations via linear translators

We show that many infinite classes of permutations over finite fields can be constructed via translators with a large choice of parameters. We first characterize some functions having linear translators, based on which several families of permutations are then derived. Extending the results of [9], we give in several cases the compositional inverse of these permutations. The connection with complete permutations is also utilized to provide further infinite classes of permutations. Moreover, we propose new tools to study permutations of the form $x\mapsto x+{(x^{p^m}-x+\delta )}^s$ and a few infinite classes of permutations of this form are proposed.     

Self-inverse Sheffer sequences and Riordan involutions

In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group.     

An involution on increasing trees

We introduce a method to construct bijections on increasing trees. Using this method, we construct an involution on increasing trees, from which we obtain the equidistribution of the statistics ‘number of odd vertices’ and ‘number of even vertices at odd levels’. As an application, we deduce that the expected value of the number of even vertices is twice the expected value of the number of odd vertices in a random recursive tree of given size.     

不动点集为RP(4)∪ P(4,2n-1)的对合

设(M,T)是一个光滑闭流形上的对合,不动点集为F=RP(4)UP(4,2n-1),则它的每一个对合(M,T)必协边(RP(4)×RP(4),twist)和(P(4,2n),T')之一.