Unitriangular matrix group

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Definition

Definition as matrix group

Suppose R is a commutative unital ring and n is a natural number. The unitriangular matrix group, denoted U(n,R), UT(n,R), or UL(n,R), is the group, under multiplication, with 1s on the diagonal, 0s below the diagonal, and arbitrary entries above the diagonal.

Note that the symbol U is also used for the unitary group, hence we use UT or ULto avoid confusion.

Explicitly:

UT(n,R) = \left \{ \begin{pmatrix} 1 & * & * & \dots & * \\ 0 & 1 & * & \dots & * \\ 0 & 0 & 1 & \dots & * \\ \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & 1 \\\end{pmatrix} \mid \mbox{all star-marked entries vary arbitrarily over } R \right \}

Definition by presentation

The presentation given here is similar to the presentation used for the Steinberg group over a unital ring. Specifically, we use only those generators and relations that correspond to upper triangular matrices and discard the rest. We denote by e_{ij}(\lambda) the matrix with 1s on the diagonal, \lambda in the (i,j)^{th} entry, and zeros elsewhere. Then:

  • The following is a generating set: For every element \lambda \in R and for 1 \le i <j \le n, include the element e_{ij}(\lambda).
  • We can get a presentation of the group using this generating set, by including the following relations:
    • e_{ij}(\lambda)e_{ij}(\mu) = e_{ij}(\lambda + \mu) (note that this in particular implies that e_{ij}(0) is the identity element for all i,j).
    • [e_{ij}(\lambda),e_{jk}(\mu)] = e_{ik}(\lambda \mu) (the Steinberg group version also imposes the condition i \ne k, but this constraint follows automatically in the upper triangular case because i < j and j < k).
    • [e_{ij}(\lambda),e_{kl}(\mu)] = 1 (i.e., is the identity element) for i \ne l, j \ne k.

Note that this presentation can be trimmed quite a bit. In fact, if A is a generating set for the additive group of R, the set:

\{ e_{i,i+1}(\lambda) \mid 1 \le i \le n - 1, \lambda \in A \}

is a generating set for UT(n,R), and we can work out a presentation in terms of this generating set using the relations above.

Case of a field

When F is a field, the unitriangular matrix group UT(n,F) can also be described as a maximal unipotent subgroup of the general linear group GL(n,F). It is also a maximal unipotent subgroup of the special linear group SL(n,F).

Case of a finite field

When F is a finite field with q elements and characteristic p (so q is a power of p), then UT(n,F) is also denoted UT(n,q), and is a p-Sylow subgroup of GL(n,F) = GL(n.q).

Particular cases

Value of n n(n-1)/2 (the "dimension" of UT(n,R), the size is |R|^{n(n-1)/2}) Nature of unitriangular matrix group UT(n,R) over a ring R
1 0 trivial group
2 1 isomorphic to the additive group of R
3 3 see unitriangular matrix group of degree three
4 6 see unitriangular matrix group of degree four

Arithmetic functions

We give here the arithmetic functions for UT(n,R).

Function Value Similar groups Explanation
order |R|^{n(n-1)/2}
nilpotency class n - 1
derived length \lceil \log_2n \rceil (the smallest integer greater than or equal to \log_2 n
Frattini length PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] (depends both on the nature of R and on n)