Tensor product of two unipotent matrices Let $u$ and $v$ be unipotent matrices with entries from some field $k$, wlog assume they are full Jordan blocks. What is known about the Jordan decomposition of the tensor product of $u$ and $v$? In characteristic 0 there is a simple formula which no longer holds in positive characteristic. Does it hold eg under some restrictions on the sizes of $u$ and $v$?
EDIT
To be more precise, a unipotent $n \times n$ matrix has its Jordan form determined by the sizes of the Jordan blocks, ie. by a partition $a_1, \ldots, a_k$ of $n$ (here $\sum a_i = n$).
Then in the characteristic $0$ case we know that if $u$ is a $p \times p$ Jordan block and $v$ is a $q \times q$ Jordan block (say $p \geq q$), then $u \otimes v$ corresponds to the partition $a_1, \ldots, a_q$, where $a_i = p + q - (2i - 1)$. This is no longer true in positive characteristic.
 A: Here is the easy partial answer. If you fix the size of the two blocks then for large enough characteristic the decomposition is the same as it is in characteristic 0.
The cheap way is to just realize that the existence of a matrix conjugating one given matrix (in your case the tensor product of the two jordan blocks) into another given matrix (the usual decomposition in characteristic 0) can be formulated into a statement in first order logic.  So Robinson's principle (which is just a compactness argument) tells us that any first order statement in the language of fields that is true over $\bar{\mathbb{Q}}$ is true over the algebraic closure of $\mathbb{F}_p$ for all sufficiently large $p$.
Of course that sort of argument sheds little light on just how large you need to make the characteristic for this to hold.  In fact, we only need the characteristic to be larger than the largest "expected" size of the Jordan blocks in the tensor product, but this doesn't have as easy of an explanation.
Understanding what happens for small characteristic is more subtle.  If I recall correctly, there is no simple formula but instead a recursive procedure for such decompositions. This problem is closely related to the representation theory of $SL(2)$ in positive characteristic. To understand that in more detail I think a good place to start is Jantzen's book "Representations of algebraic groups".
