Testing Membership in Unitriangular Matrix Groups Preliminary Draft Gg Abor Ivanyos

The Width of Verbal Subgroups in the Group of Unitriangular Matrices over a Field

Let K be a field and let UTn(K) and Tn(K) denote the groups of all unitriangular and triangular matrices over field K, respectively. In the paper, the lattices of verbal subgroups of these groups are characterized. Consequently the equalities between certain verbal subgroups and their verbal width are determined. The considerations bring a series of verbal subgroups with exactly known finite wi...

Hopf monoids from class functionson unitriangular matrices

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatori...

Unitriangular factorisations of Chevalley groups

Computing in Solvable Matrix Groups

We announce methods for e cient management of solvable matrix groups over nite elds. We show that solvability and nilpotence can be tested in polynomial-time. Such e ciency seems unlikely for membership-testing, which subsumes the discrete-log problem. However, assuming that the primes in jGj (other than the eld characteristic) are polynomiallybounded, membership-testing and many other computat...

Commutators and powers of infinite unitriangular matrices

In the paper we consider some commutator-type and power-type matrix equations in the group UT(∞,K) of infinite dimensional unitriangular matrices over a fieldK. We introduce a notion of a power outer commutator ω1k k (x1, ..., xk) and a power Engel commutator e1k k (x, y) as outer (respectively Engel) commutators modified by allowing powers of letters instead of letters alone. For a given infin...