Monoidal categories for the combinatorics of group representations

Categories are known to be useful for organizing structural aspects of mathematics. However, they are also useful in finding out what structure can be dismissed (coherence theorems) and hence in aiding calculations. We want to illustrate this for finite set theory, linear algebra, and group representation theory. We begin with some combinatorial set theory. Let N denote the set of natural numbers. We identify each n⁄⁄Œ⁄⁄N with the finite set n = { j⁄⁄Œ⁄⁄N: 0 £ j < n}. However, we must be careful to distinguish the cartesian product m ⁄⁄ ¥ ⁄⁄n = { (i⁄⁄,⁄⁄j) : 0 £ i < m, 0 £ j < n} from the isomorphic set mn. Let S denote the skeletal category of finite sets; explicitly, the objects are the n⁄⁄Œ⁄⁄N and the morphisms are the functions between these sets. We need to discuss the explicit construction of finite products in S⁄⁄. Let p 0 m n, : mn aAm and p1 m n, : mn aAn be the functions given by p 0 m n, (k) = i and p1 m n, (k) = j where k = i ⁄⁄n + j. That p 0 m n, and p1 m n, are functions, and that (mn⁄⁄, ⁄ p 0 m n,,⁄ ⁄ p1 m n, ) is the categorical binary product of m and n in S⁄⁄, follow from the fact that each natural number 0 £ k < mn has a unique expression i n the form k = i ⁄⁄n + j with 0 £ i < m and 0 £ j < n. Moreover, each natural number 0 £ h < mnp has a unique expression h = i ⁄⁄n ⁄⁄p + j ⁄⁄p + k with 0 £ i < m ⁄, 0 £ j < n and 0 £ k < p. So, referring to the diagram mnp mn m n p mnp np m n p p1 mn p, p 0 mn p, p 0 m n, p 0 m np, p 1 m np, p 1 m n, p 1 n p, p 0 n p, we see that p 0 m n, ∞ ⁄ p 0 mn p, = p 0 m np, , p 1 m n, ∞ ⁄ p 0 mn p, = = p 0 n p, ∞ ⁄ p 1 m np, , p 1 mn p, = p 1 n p, ∞ ⁄ p 1 m np,. We denote these last three functions by p 0 m n p, , : mnp aAm ⁄⁄, p 1 m n p, , : mnp aAn ⁄⁄, p 2 m n p, , : mnp aAp⁄⁄, to obtain a categorical ternary product (mnp⁄⁄, p