Reaction between potassium permanganate, alum, and sodium hydrogen sulfite
The fact that the solution turned clear is a pretty good guarantee that this was a redox reaction: the permanganate oxidised some other species, and in turn was reduced, probably to manganese dioxide, the brown precipitate you noticed. Unfortunately, neither of the other reactants you mention are reducing agents!
Do you think it's possible that the third reagent was actually sodium hydrogen sulphite, NaHSO$_3$? If so, this would make a lot more sense: the (hydrogen)sulfite ion would have been oxidised to (hydrogen)sulfate. This reaction requires an acidic medium, and the reason the alum (potassium aluminium sulfate) was necessary was to lower the pH.
These individual steps have the following equations:

Al$^{3+}$ + 4 H$_2$O $\rightarrow$ [Al(OH)$_4$]$^-$ + 4 H$^+$
MnO$_4^-$ + 4 H$^+$ + 3 $e^-$ $\rightarrow$ MnO$_2\downarrow$ + 2 H$_2$O
HSO$_3^-$ + H$_2$O $\rightarrow$ HSO$_4^-$ + 2 H$^+$ + 2 $e^-$

(note that the last two are half-equations -- they can’t occur on their own, but I’ve written them separately for clarity) giving an overall reaction of

Al$^{3+}$  + 2 MnO$_4^-$ + 3 HSO$_3^-$ + 3 H$_2$O $\rightarrow$ [Al(OH)$_4$]$^-$ + 2 MnO$_2\downarrow$ + 3 HSO$_4^-$

The white crystalline deposit in this case would just be alum again, more or less. You have some sodium in your solution now, from the sodium hydrogensulfite, so you'd get some sodium aluminium sulfate as well, but the only anion in your solution will still be (hydrogen)sulfate, and my guess is that with aluminium this will crystallise out as the sulfate.