What's the advantage of using array programming versus using list/array types in other languages? Nothing I saw in this article seems particularly unique to J except the syntax.
Probably the built in support for vectorization, which reduces the amount of code you need to write, and allows your code to be closer to mathematical expressions. Of course there are other languages like R, Julia and Fortran that also have built-in vectorization, and Python adds that with the Numpy library. It makes certain operations much easier to express than with your normal lists/arrays. That's why the Python scientific stack is built on top of Numpy arrays, and not the builtin data structures.
You can compare it to regular expressions for manipulating text.
This setup often gives APL-family languages a curious performance advantage, even though they’re largely dynamic & interpreted. All of the hot loops are in the implementation layer, and the surface language just glues them together, just like how Python can “be” fast by acting as glue for C and Fortran.
Parallelization automatically, special constructs for sparse matrixes, complex manipulation of multidimensional arrays without need for making sure they are constructed correctly. Not to mention never having to worry about an indexing error, unless you are specifically asking for an index, in which case the bug is in your design, not the program. Built in methods for sorting and searching that are very efficient, as well as filtering. Also, J is faster when you give it all the data you can at once, rather than moving through index by index. A lot of these advantages take time to really be aware of, much like Erlang's bread and butter often hiding behind much more than one article.
I'd take this a step further and say the website for J language doesn't move a finger to entice anyone to learn the language or use it.
I don't know if it's academia or what - but if I spent the effort on creating a whole programming language that I believed was actually good - I'd have a very different front page.
Although some academics (like myself) are intrigued by them, APL-like languages (APL, J, K, Q, etc.) have very little to do with academia historically, and were developed pretty much isolated from the academic programming languages community. APL itself does have distant roots in academia, growing out of notation Iverson developed at Harvard in the 1950s, but its history as a programming language goes via IBM and a variety of other companies generally focused on enterprise and finance (I.P. Sharp, Morgan Stanley, Dyalog, Kx Systems).
As for J, it was released by a startup, J Software, founded by Kenneth Iverson and Roger Hui in 1990 for that purpose. It was later open sourced in 2011.
It was probably fine in the 90's when I'm guessing it was written. I personally think the minimalistic design is great, which is a trait of both the language and community. The problem is different parts of the site have different and inconsistent menus, so navigation can be cumbersome at times.
You can compare it to regular expressions for manipulating text.