Free group of infinite unitriangular matrices

In the group of infinite unitriangular matrices over the field with two elements, a free subgroup of rank two is constructed which is a group of finite-automata transformations over a two-element alphabet. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 382–386, March, 2000.     

On the groups of unitriangular automorphisms of relatively free groups

We describe the structure of the group U _n of unitriangular automorphisms of the relatively free group G _n of finite rank n in an arbitrary variety C of groups. This enables us to introduce an effective concept of normal form for the elements and present U _n by using generators and defining relations. The cases n = 1, 2 are obvious: U ₁ is trivial, and U ₂ is cyclic. For n ?? 3 we prove the following: If G _n?1 is a nilpotent group then so is U _n . If G _n?1 is a nilpotent-by-finite group then U _n admits a faithful matrix representation. But if the variety C is different from the variety of all groups and G _n?1 is not nilpotent-by-finite then U _n admits no faithful matrix representation over any field. Thus, we exhaustively classify linearity for the groups of unitriangular automorphisms of finite rank relatively free groups in proper varieties of groups, which complements the results of Olshanskii on the linearity of the full automorphism groups AutG _n . Moreover, we introduce the concept of length of an automorphism of an arbitrary relatively free group G _n and estimate the length of the inverse automorphism in the case that it is unitriangular.     

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let Un(Fq) denote the group of unipotent n×n upper triangular matrices over a finite field with q elements. We show that the Heisenberg characters of Un+1(Fq) are indexed by lattice paths from the origin to the line x+y=n using the steps (1,0), (1,1), (0,1), (0,2), which are labeled in a certain way by nonzero elements of Fq. In particular, we prove for n?1 that the number of Heisenberg characters of Un+1(Fq) is a polynomial in q−1 with nonnegative integer coefficients and degree n, whose leading coefficient is the nth Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of Un(Fq) is a polynomial in q−1 whose coefficients are Delannoy numbers and whose values give a q-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of Un(Fq) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in q−1 with nonnegative integer coefficients.     

Counting characters of small degree in upper unitriangular groups

Let U_n denote the group of upper n×n unitriangular matrices over a fixed finite field of order q. That is, U_n consists of upper triangular n×n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U_n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U_n of degree q^e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1,2 and 3, and thus show that the conjecture is true in those cases.     

Decomposition of infinite matrices

We consider the problem of decomposing a matrix of integers according to constraints on the row and column sums, in the case when the original matrix is infinite. Some necessary conditions and some sufficient conditions are found for the existence of such a decomposition, generalizing results of Ball for the finite case. Connections with the Transversal Problem for infinite sets are pointed out.     

Subgroups of powerful groups

We consider subgroups of powerfulp-groups. In particular, we give a new proof that allp-groups are sections of powerfulp-groups, give necessary and sufficient conditions for a 2-generator group to be a normal subgroup of a powerfulp-groups, and show thatp-groups of class 2, orp-groups with a cyclic commutator subgroup, are such normal subgroups.     

Subgroups of free groups

With each subgroup A of a free group F there is associated a number FA called the quasi-index. It is proved that FA=ªA¦ if ¦FA¦ is finite. Some properties of the quasi-index are also established, in particular that the analog of Lagrange's theorem is valid for it: FB FA AB if A B as well as generalizations of Howson's and Byrnes' theorems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1115–1119, July–August, 1991.     

Eigenvalues for infinite matrices

This paper is concerned with the problem of determining the location of eigenvalues for diagonally dominant infinite matrices; upper and lower bounds for eigenvalues are established. For tridiagonal matrices, a numerical procedure for improving the bounds is given, and the approximation of the eigenvectors is also discussed. The techniques are illustrated for the solution of the well-known Mathieu's equation.     

Diagonalization of row-column-finite infinite matrices

A complete solution to the diagonalization problem (under equivalence) for row-column-finite infinite matrices over a general field is given Project supported by the National Natural Science Foundation of China