Let $f(x_1,...,x_n)$ be a complex polynomial. Show the following two conditions on $f$ are equivalent: i) for any transpositions $\tau$ we have $\tau.f=-f$ and ii) for any $\sigma \in S_n$ we have $\sigma f=f$ when $\sigma$ is even and $\sigma .f=-f$ when $\sigma$ is odd. Such a polynomial is said to be anti-symmetric. Show that $\Delta (x_1,...x_n)$ is anti symmetric, where $\Delta (x_1,...x_n)$=$\Pi_{i<j}(x_i-x_j)$.

I should also note that $\tau.f=f(x_{\tau(1)},...x_{\tau(n)})$. I'm unsure how to define the function so I can start the proof. Furthermore for the last part the difference function is given as $(x_n-x_{n-1})(x_n-x_{n-2})....(x_2-x_1)$. I'm trying to find a relationship between the number of times a "swap" will occur upon a transposition and furthermore conclude that it's antisymmetric. I understand why $\Delta (x_1,...x_n)^2$ is symmetric though. Thank you in advanced.


1 Answer 1


The transpositions $s_i = (i \; i+1) \in S_n$, $(1 \le i < n)$ generate and satisfy the Coxeter relations $$ \begin{align} s_i^2 &= 1 \\ s_i s_j &= s_j s_i \quad &\text{if } |i - j| > 1 \\ s_i s_j s_i &= s_j s_i s_j \quad &\text{if } |i - j| = 1 \end{align} $$ In other words, any permutation $\tau \in S_n$ can be written (not uniquely) as a product of these standard generators. $$ \tau = s_{i_1} s_{i_2} \cdots s_{i_r} $$ The bubble sort algorithm implements this.

The length $\ell(\tau)$ of a permutation $\tau$ is the minimum length of all words that yield that permutation.

(Try to verify for yourself that $\ell(\tau) = 3$ for $\tau = (1 \; 3) \in S_3$.)

Now, the sign $\operatorname{sgn}(\tau) = (-1)^{\ell(\tau)}$, and a permutation is called even (resp., odd) if the length of the permutation is even (resp., odd).

To see that the Vandermonde polynomial $\Delta(x_1, \dots, x_n)$ is alternating (anti-symmetric), observe that a single transposition, say $(i \; j)$ has the following effect on the factors $(x_t - x_u)$: $$ \begin{align} (x_i - x_u) &\longleftrightarrow (x_j - x_u) & \text{if } t \in \{i, j\} \text{ and } u \notin \{i, j\} \\ (x_t - x_i) &\longleftrightarrow (x_t - x_j) & \text{if } t \notin \{i, j\} \text{ and } u \in \{i, j\} \\ (x_i - x_j) &\longleftrightarrow (x_j - x_i) & \text{if } \{s, t\} = \{i, j\} \end{align} $$ All other factors are fixed. In the first two cases, two factors in the polynomial algebra are interchanged, so the product is unchanged. The last case involves replacing a factor by its opposite, which switches the sign of $\Delta$.


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