# Abstract Nonsense

## Symmetric and Skew-Symmetric forms

Point of post: In this post I discuss the equivalent of section 29 and the end of 30, in more detail than is done in Halmos’s book.

Motivation

In my the last post we discussed how one can extend the notion of bilinear forms; one of the main reasons being because we desired to extend the notion of a determinant to higher dimensions (in some intuitive sense). It turns out though that the generality to which we extended it is often (not always) too general for the level of linear algebra we are operating at. In particular, instead of studying $\text{Mult}\left(\mathscr{V}_1,\cdots,\mathscr{V}_n\right)$ when $\mathscr{V}_1,\cdots,\mathscr{V}_n$ are any arbitrary, possibly distinct vector spaces over some field $F$ we instead intend to study $\text{Mult}\left(\mathscr{V}_1,\cdots,\mathscr{V}_n\right)$ where $\mathscr{V}_1=\cdots=\mathscr{V}_n=\mathscr{V}$. In this case we denote $\text{Mult}\left(\mathscr{V}_1,\cdots,\mathscr{V}_n\right)$ by $\text{Mult}_n\left(\mathscr{V}\right)$. The reason why we would want to consider this scenario is because when we restrict our attention to multilinear forms on the direct sum of the same vector space the resultant space has some beautiful interplay with $S_n$. In other words,  there is a very natural way in which $S_n$ interacts with the elements of $\text{Mult}_n\left(\mathscr{V}\right)$. Our first indication of this was symmetric bilinear forms. Which, phrased slightly differently say that $B(x_1,x_2)=B(x_{\pi(1)},x_{\pi(2)})$ for every $\pi\in S_2$, which isn’t very profound considering there are only two elements of $S_n$. But, we will be able to go further and say, in a precise manner yet to be said, that $S_n$ acts on $\text{Mult}_n\left(\mathscr{V}\right)$ so that there is a meaningful way to “multiply” elements of $S_n$ and elements of $\text{Mult}_n\left(\mathscr{V}\right)$. This, is where all our work with permutations starts to come into play (though, to be quite honest I probably developed the theory a little further than it really needed to be).

Remark: Halmos defines something in his book either incorrectly, or ascribes to it a property which isn’t true (see below).

$S_n$‘s Action On $\text{Mult}_n\left(\mathscr{V}\right)$

Let $\mathscr{V}$ be a finite dimensional vector space over $F$. Then, for every $\pi\in S_n$ and every $K\in\text{Mult}_n\left(\mathscr{V}\right)$ there is a way to associate the two to create a new element $\pi K$ of $\text{Mult}_n\left(\mathscr{V}\right)$ given by $(\pi \cdot K)\left(x_1,\cdots,x_n\right)=K\left(x_{\pi^{-1}(1)},\cdots,x_{\pi^{-1}(n)}\right)$. For example, if

$\pi=\begin{pmatrix}1 & 2 & 3 & 4\\4 & 2 & 1 & 3\end{pmatrix}$

then $(\pi \cdot K)(1,3,9,2)=K(2,3,1,9)$. Indeed this is a $n$-form since

\begin{aligned}\left(\pi\cdot K\right)\left(x_1,\cdots,\alpha x_k+\beta x'_k,\cdots,x_n\right) &= K\left(x_{\pi^{-1}(1)},\cdots,\alpha x_{\pi^{-1}(k)}+\beta x'_{\pi^{-1}(k)},\cdots,x_{\pi^{-1}(n)}\right)\\ &= \alpha K\left(x_{\pi^{-1}(1)},\cdots,x_{\pi^{-1}(k)},\cdots,x_{\pi^{-1}(n)}\right)+\beta K\left(x_{\pi^{-1}(1)},\cdots,x'_{\pi^{-1}(k)},\cdots,x_{\pi^{-1}(n)}\right)\\ &=\alpha\left(\pi\cdot K\right)\left(x_1,\cdots,x_k,\cdots,x_n\right)+\beta\left(\pi\cdot K\right)\left(x_1,\cdots,x'_k,\cdots,x_n\right)\end{aligned}

So, with this we can state the main reason for restricting our attention to $\text{Mult}_n\left(\mathscr{V}\right)$:

Theorem: Let $\mathscr{V}$ be a finite dimensional $F$-space, then $S_n$ acts on $\text{Mult}_n\left(\mathscr{V}\right)$ via the action

$\cdot:S_n\times\text{Mult}_n\left(\mathscr{V}\right)\to\text{Mult}_n\left(\mathscr{V}\right):\left(\pi,K\right)\mapsto \pi\cdot K$

where

$\displaystyle \left(\pi\cdot K\right)(x_1,\cdots,x_n)=K\left(x_{\pi^{-1}(1)},\cdots,x_{\pi^{-1}(n)}\right)$

Proof: It suffices to prove that $\text{id}_{[n]}\times K=K$ for all $K\in\text{Mult}_n\left(\mathscr{V}\right)$ and

$\sigma\cdot\left(\tau\cdot K\right)=\left(\sigma\tau\right)\cdot K$

for all $\sigma,\tau\in S_n$. So, the first of these is clear since

$\left(\text{id}_{[n]}\cdot K\right)\left(x_1,\cdots,x_n\right)=K\left(x_{\text{id}^{-1}{[n]}},\cdots,x_{\text{id}^{-1}_{[n]}(n)}\right)=K(x_1,\cdots,x_n)$

for all $x_1,\cdots,x_n\in\mathscr{V}$.  For the second condition we merely note that

\begin{aligned}\left(\sigma\cdot\left(\pi\cdot K\right)\right) &=\left(\pi\cdot K\right)\left(x_{\sigma^{-1}(1)},\cdots,x_{\sigma^{-1}(n)}\right)\\ &= K\left(x_{\pi^{-1}\left(\sigma^{-1}\left(1\right)\right)},\cdots,x_{\pi^{-1}\left(\sigma^{-1}\left(n\right)\right)}\right)\\ &= K\left(x_{(\sigma\pi)^{-1}(1)},\cdots,x_{(\sigma\pi)^{-1}(n)}\right)\\ &= \left(\left(\sigma\pi\right)\cdot K\right)\left(x_1,\cdots,x_n\right)\end{aligned}

and since $x_1,\cdots,x_n\in \mathscr{V}$ were arbitrary it follows that $\sigma\cdot\left(\pi\cdot K\right)=\left(\sigma\pi\right)\cdot K$ as required.  $\blacksquare$

From hereon out we drop the $\cdot$ in $\pi\cdot K$ in lieu of $\pi K$.

We are now prepared to discuss two important classes of $n$-forms whose definitions are precipitated by the above theorem. Namely the generalization of symmetric bilinear forms t and their counterparts skew-symmetric bilinear forms. More explicitly, we call $K\in\text{Mult}_n\left(\mathscr{V}\right)$ symmetric if $\pi K=K$ for all $\pi\in S_n$ (equivalently that the isotropy subgroup generated by $K$ is the full group, for those  who know the jargon) and we call $K$ skew-symmetric if $\pi K=\text{sgn}(\pi)K$ for all $\pi\in S_n$.

Now, with this it’s clear that not all $n$-forms are symmetric, nor are all skew-symmetric, in fact not all are either. That said, we will see that from each $n$-linear form $K$ there is a natural way for one to build a symmetric and an skew-symmetric form. Namely:

Theorem: Let $\mathscr{V}$ be a finite dimensional $F$-space and $K\in\text{Mult}_n\left(\mathscr{V}\right)$ then the $n$-form $\text{Sym}\left(K\right)$ given by

$\displaystyle \text{Sym}\left(K\right)=\sum_{\pi\in S_n}\pi K$

is symmetric.

Proof: Let $\sigma\in S_n$ be fixed but arbitrary and let $\sigma S_n=\left\{\sigma \pi:\pi\in S_n\right\}$. We claim that $\sigma S_n=S_n$. To do this it suffices to show that

$f_\sigma:S_n\to S_n:\pi\mapsto \sigma\pi$

is a bijection. To prove injectivity we merely recall that $\sigma\pi=\sigma\pi'\implies \pi=\pi'$ and surjectivity follows from $f_{\sigma}\left(\sigma^{-1}\pi\right)=\pi$ for all $\pi\in S_n$ from where bijectivity follows( this is really just a statement of Cayley’s theorem for $S_n$). Thus, we see that

$\displaystyle \sigma\text{Sym}\left(K\right)=\sigma\sum_{\pi\in S_n}\pi K=\sum_{\pi\in S_n}\sigma\pi K=\sum_{\tau\in \sigma S_n}\tau K=\sum_{\tau\in S_n}\tau K=\text{Sym}\left(K\right)$

from where it follows by the arbitrariness of $\sigma$ that $\text{Sym}\left(K\right)$ is symmetric. $\blacksquare$

As would be expected there is a analogous way to naturally procure an skew-symmetric form from any $n$-form $K$, namely:

Theorem: Let $\mathscr{V}$ be a finite dimensional $F$-space and $K\in\text{Mult}_n\left(\mathscr{V}\right)$. Then, the $n$-form $\text{Asym}\left(K\right)$ given by

$\displaystyle \text{Asym}\left(K\right)=\sum_{\pi\in S_n}\text{sgn}(\pi)\pi K$

is skew-symmetric.

Proof: Let $\sigma\in S_n$ be arbitrary. Using the fact that $\sigma S_n=S_n$ as proven in the last theorem we see that

\displaystyle \begin{aligned}\sigma\text{Asym}\left(K\right) &=\sigma\sum_{\pi\in S_n}\text{sgn}(\pi)\pi K\\ &=\sum_{\pi\in S_n}\frac{1}{\text{sgn}(\sigma)}\text{sgn}\left(\sigma\pi\right)\sigma\pi K\\ &=\text{sgn}(\sigma)\sum_{\tau \in \sigma S_n}\text{sgn}(\tau)\tau K\\ &=\text{sgn}(\sigma)\sum_{\tau\in S_n}\text{sgn}(\tau)\tau K\\ &=\text{sgn}(\sigma)\text{Asym}\left(K\right)\end{aligned}

where we’ve made tacit use of the fact that $\text{sgn}(\sigma\pi)=\text{sgn}(\sigma)\text{sgn}(\pi)$ (proof can be be found here) and that $\displaystyle \frac{1}{\text{sgn}(\sigma)}=\text{sgn}(\sigma)$ (since $\text{sgn}(\sigma)=\pm 1$). Thus, since $\sigma$ was arbitrary the conclusion follows. $\blacksquare$

Remark: $\text{Asym}$ is for antisymmetric another name for skew-symmetric forms.

References:

1. Halmos, Paul R. ” Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print

November 14, 2010 -