Background:
Coming from a non-field theoretic background I was recently thinking about symmetric $n$-ary scalar products as generalizations of inner products. Roughly, I was asking myself the following question:
- if an inner product $\langle x,y \rangle$ gives you a notion of angle between two vectors $x,y$ in the form of $\cos(\theta)= \frac{\langle x,y \rangle}{|x||y|}$, what additional conditions do we need to impose on a symmetric 3-ary scalar product $\langle x,y,z \rangle$ to be able to define a similar notion for the three vectors $x, y$ and $z$ (if possible, of course).
So far so naive.
Instead of ordinary symmetry, for inner products over complex numbers we actually have the requirement of conjugate symmetry:
$$\overline{\langle x,y \rangle} = \langle y,x \rangle.$$
Leaving the case of vector spaces for later, I thought it might therefore be a good idea to first generalize the product $\langle x,y \rangle =x\bar{y}$ in $\mathbb{C}$ itself to $n$-ary products over other fields.
Main part:
To this end I identified complex conjugation as the action of the generator $\tau \in \operatorname{Gal}(\mathbb{C}/\mathbb{R}) \cong S_2$.
It is well-known that the Galois group $\operatorname{Gal}(L/K)$ of the splitting field $L$ of a polynomial $f$ of degree $n$ over a field $K$ is a subgroup of $S_n$. Therefore, I thought it would be natural to generalize the condition of symmetry for an $n$-ary product in $L/K$ as follows: $$\sigma.\langle x_1, \dots,x_n \rangle = \langle x_{\sigma(1)}, \dots, x_{\sigma(n)}\rangle \qquad \forall \sigma \in \operatorname{Gal}(L/K) \leq S_n.$$ Among other nice features this has the immediate consequence (at least in characteristic 0) that if $ \sigma \in \operatorname{Gal}(L/K) $ partitions the set $\{1,\dots,n \}$ into cycles so that the $x_i$'s with indeces in the same cycles are equal, then $\langle x_1, \dots,x_n \rangle \in L^{\langle \sigma \rangle}$. Compare this to $\langle x,x \rangle \in \mathbb{R}$ in the complex case.
Next, I wanted to see if (a) such a product always exists and (b) if there is a canonical (in some sense) choice.
I looked at the well-known example of $f(x)=x^3-2$ over $\mathbb{Q}$. Let $X=\sqrt[3]{2}$ and let $Y$ be a primitive third root of unity. Moreover, let $\alpha$ be a generator of the cyclic subgroup of order 3 in $\operatorname{Gal}(\mathbb{Q}(X,Y)/\mathbb{Q})$ that fixes $\mathbb{Q}(Y)$ and let $\tau$ be the generator of the cyclic subgroup that fixes $\mathbb{Q}(XY)$. I set $$ [x,y,z]:= x \cdot(\alpha.y) \cdot (\alpha^2.z). $$ This is already not bad as the above symmetry condition is satisfied for all cyclic permutations. In order to get also the 'dark half' of $S_3$, I set: $$ \langle x,y,z \rangle := \frac{[x,y,z]+\tau.[z,y,x]}{2}. $$ This definition satisfies the full symmetry condition above. Moreover, we can pull out
- $x$, when $x \in \mathbb{Q}(X)$
- $y$, when $y \in \mathbb{Q}(XY)$
- $z$, when $z \in \mathbb{Q}(XY^2).$
The above definition may seem a bit random but it was basically the only thing I could think of that worked (which doesn't mean a lot).
Question: Generally, I would like to know if people looked at stuff like this, especially the symmetric $n$-ary product part. If so, do you know in which contexts they exist and if there is a unique (up to blah) choice? Also, if you have any comment or thought I am happy to hear them!
Remark: In case you wonder, I am not doing math professionally anymore so don't judge me for fooling around like this ;)
Interesting question! I have been thinking about this for the past couple of years as part of my project on higher-arity mathematics:
https://www.arity.science/
There you can see some recent papers where operations of multi-index contraction effectively define higher-arity analogues of the dot product. The most documented case of such a multi-index contraction is the Bhattacharya-Mesner product of ternary matrices/relations. That operation has been known since the 1990s but very little research exists on it; notably E. Gnang has some papers on it.
The ternary dot product that appears in the Bhattacharya-Mesner product turns out to be the most straightforward generalization of the ordinary (binary) dot product: let $x,y,z\in \mathbb{F}^n$ for some field $(\mathbb{F},+,\cdot)$, then the tridot product is given by:
$$ \bullet (x,y,z):= \sum_{i=1}^n x_i\cdot y_i \cdot z_i. $$
This is a symmetric $\mathbb{F}$-trilinear "form". Quotation marks since it is not clear if $\bullet$ has a non-trivial symmetry group of $\text{GL}(\mathbb{F},n)$ transformations or if, indeed, the notion of symmetry group applies to such a ternary operation (it seems plausible that an appropriate ternary analogue of a symmetry transformation is required).
This is an exciting area of research but I am afraid very little is known - which is a good thing for those of us who are delving into it!