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A map of the 24 permutations and the 23 swaps used in Heap's algorithm permuting the four letters A (amber), B (blue), C (cyan) and D (dark red) Wheel diagram of all permutations of length = generated by Heap's algorithm, where each permutation is color-coded (1=blue, 2=green, 3=yellow, 4=red).
Now the character of this representation is defined as the trace of this permutation matrix. An element on the diagonal of a permutation matrix is 1 if the point in is fixed, and 0 otherwise. So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of .
Furthermore, a class function on is a character of if and only if it can be written as a linear combination of the distinct irreducible characters with non-negative integer coefficients: if is a class function on such that = + + where non-negative integers, then is the character of the direct sum of the representations corresponding to .
The following figure shows the output of all three aforementioned algorithms for generating all permutations of length =, and of six additional algorithms described in the literature. Ordering of all permutations of length = generated by different algorithms.
When θ is the trivial character of H, the induced character obtained is known as the permutation character of G (on the cosets of H). The general technique of character induction and later refinements found numerous applications in finite group theory and elsewhere in mathematics, in the hands of mathematicians such as Emil Artin , Richard ...
The signature defines the alternating character of the symmetric group S n. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (ε σ), which is defined for all maps from X to X, and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as sgn(σ) = (−1) N(σ)
For instance, in the case of n = 2, the superpermutation 1221 contains all possible permutations (12 and 21), but the shorter string 121 also contains both permutations. It has been shown that for 1 ≤ n ≤ 5, the smallest superpermutation on n symbols has length 1! + 2! + … + n! (sequence A180632 in the OEIS). The first four smallest ...
To each irreducible representation ρ we can associate an irreducible character, χ ρ. To compute χ ρ (π) where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule. [3] Note that χ ρ is constant on conjugacy classes, that is, χ ρ (π) = χ ρ (σ −1 πσ) for all permutations σ.