Okay then, here's another. 2-Hilbert spaces as a categorification of Hilbert spaces, and the <a href="http://ncatlab.org/nlab/show/Gram-Schmidt+process#categorified_gramschmidt_process_6">categorified Gram-Schmidt process</a> (which I first learned from James Dolan). 

This may be used to derive a $\mathbb{Z}$-linear basis for the representation ring of $S_n$ that consists of <i>permutation representations</i>, hence a combinatorial alternative to the basis consisting of irreducible representations. The reference above sketches how this works in the case $Rep(S_4)$.