# Upper triangular matrix homomorphisms

Let $A_n$ be the group of n-dimensional invertible upper triangular matrices over an arbitrary field.

Is there an easy way to write down $n$ different group homomorphisms $f: A_n \rightarrow A_{n+1}$?

I only found two so far: Adding a one in the additional entry $(n+1,n+1)$ or mapping any matrix to the identity.

You can add an additional $1$ anywhere on the diagonal by also adding zero's in the rest of that row and column. So for example: $$\begin{bmatrix}a & b & c \\ 0 & d & e \\ 0 & 0 & f\end{bmatrix} \mapsto \begin{bmatrix}a & 0 & b & c \\ 0 & 1 & 0 & 0 \\ 0 & 0 & d & e \\ 0 & 0 & 0 & f\end{bmatrix}.$$ All of these maps correspond to a vector space inclusion $\mathbb R^n \to \mathbb R^{n + 1}$ where one simply adds one additional basis element to the given basis. The location of the $1$ is determined by where that new basis element occurs in the ordering of the basis.

Edit: In case it's not clear, the correspondence is not taking the matrix of the inclusion $\mathbb R^n \to \mathbb R^{n + 1}$, that would not give a map of matricies, that would give a single non-square matrix. The correspondence is to consider an upper triangular $n \times n$ matrix as a map on $\mathbb R^n$, extend this map to $\mathbb R^{n + 1}$ by declaring it to be the identity on the new basis element, and then take the matrix that represents this new linear map.