TITLE: Complexity of rectangular matrix multiplication
QUESTION [5 upvotes]: I am interested in the complexity of multiplying two matrices $A$ and $B$, i.e. to compute $AB$.
From [Le Gall and Urrotia], I know that:

if $A$ and $B$ are square-matrices of size $n$, then this can be done in  $O(n^{\omega})$ where  $\omega\approx 2.372$.
if $A$ has size $n\times n^{k}$ and $B$ has size $n^k \times n$ this can be done in $O(n^{\omega(k)})$ with $\omega(k)<\omega$ for $k<1$ (typically, $\omega(k)=2$ for $k\le0.3$).

I am wondering if there an efficient algorithm when $A$ has size $n^k\times n$ and $B$ has size $n\times n$ (or when $B$ has size $n\times n^{k}$), for $k\in(0,1)$.
Remarks:

when I say "efficient" algorithm, I mean an algorithm that a complexity smaller than the naïve $O(n^{2+k})$ algorithm.  I am not refering to an actual implementation, just the existence of an algorithm.
I suppose that additing or multiplying two entries of a matrix is $O(1)$.

[Le Gall and Urrotia] Improved Rectangular Matrix Multiplication using Powers of the
Coppersmith-Winograd Tensor by François Le Gall† Florent Urrutia

REPLY [3 votes]: Thanks to Jukka's answer, I found the paper https://www.sciencedirect.com/science/article/pii/S0885064X98904769, which cites a result of http://www.mathnet.ru/links/70088989aead859688ca36af8d30e329/rm5125.pdf (unfortunately in Russian). If I understand this paper correctly (and in particular around Equation (2.7)), this paper claims that for any given exponent $r,s,t>0$, the complexity of the multiplication $<n^r,n^s,n^t>$ (i.e. multiply a $n^r\times n^s$ matrix by a $n^s\times n^t$) is independent by permutation of $r,s,t$.
This implies for my original question that the complexity of the multiplication $<n,n^k,n>$ is upper bounded by $\omega(k)$.