Every permutation can be decompose as a product of transpositions. This decomposition is not unique, however the parity of the number of transpositions that appears in the decomposition is always the same. Thus the sign is defined as $(-1)^l$ where $l$ is the number of transposition.
The sign can be calculated if the cycle decomposition if known because the sign is multiplicative and the sign of a $k$-cycle is $(-1)^(k+1)$. This is the way the method permutationSign calculates the sign.
The sign permutation is used to calculate polytabloids and higher Specht polynomials.
i1 : perm = {2,1,4,3,0} o1 = {2, 1, 4, 3, 0} o1 : List |
i2 : c = cycleDecomposition perm o2 = {{0, 2, 4}, {1}, {3}} o2 : List |
i3 : permutationSign perm o3 = 1 |
i4 : perm2 = {4,2,1,0,3} o4 = {4, 2, 1, 0, 3} o4 : List |
i5 : c2 = cycleDecomposition perm2 o5 = {{0, 4, 3}, {1, 2}} o5 : List |
i6 : permutationSign perm2 o6 = -1 |
The object permutationSign is a method function.