Concept

Irreducible

An object with no smaller one of its kind hiding inside it. A representation is irreducible when it has no invariant subspace, and the irreducible ones are the pieces every other is assembled from.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Two factor trees of 360. The same number split two different ways, both ending in the same primes.

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

number · Unique factorisation

The character table of the permutations of 4 places, computed from traces. A table with one row per irreducible representation and one column per conjugacy class, giving the trace of the matrix each representation assigns, with the dimension column marked.

When the label may be a matrix

A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.

algebra · Permutation parity

Named alongside it

The objects these essays reach for when they reach for this one.

Character table Conjugacy class Counting two ways Divisibility Euclid lemma Greatest common divisor Group representation Norm Orthogonality Permutation Primes Proof by contradiction

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