# permutationSign -- the sign of a permutation

## Synopsis

• Usage:
permutationSign(perm)
• Inputs:
• perm, a list, a permutation of the numbers from 0 to n-1
• p, an instance of the type Partition, a partition that represents the conjugacy class of the permutation
• Outputs:
• an integer, 1 or -1, the sign of the permutation

## Description

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

## Ways to use permutationSign :

• "permutationSign(List)"
• "permutationSign(Partition)"

## For the programmer

The object permutationSign is .