Concept

Sign — where it appears

The single bit a permutation carries: whether it is built from an even or an odd number of swaps. It is the only label on permutations that multiplies correctly into a commutative group, and it is what makes a determinant possible.

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

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

The crossings that will not come out even

Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

algebra · Permutation parity

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.

Permutation Bijection Character table Conjugacy class Counting argument Determinant Group Group representation Invariant Irreducible Orthogonality Parity

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