Here's a chaser to this question.

Recall the proof that the number of partition of an integer $n$ into at most $k$ addends is the same as the number of partition of an integer $n$ into integers no larger than $k$: you flip the corresponding Young's diagrams (drawn in French notation) over the main quadrant diagonal. Is this flipping operation related to a Fourier transform of something out there?

You can also add a couple of binary operations on Young diagrams: "multiplication" and "convolution". "Multiplication" draw the 1st diagram above the 2nd one and drops the columns to close the gaps: $(a*b)_i=a_i+b_i$, where $a_i,b_i$ are $i^{th}$ addends in the partitions. Similarly, "convolution" draw the 1st diagram on the right of the 2nd one and "pulls" rows to the left until the gaps are closed. Clearly, the above "flipping operation" interchanges multiplication and convolution. Is that property enough to model the "flipping operation" as a Fourier transform on something?