## Root systems and sums of squares

It is easy to see that the quadratic form for the root system $A_n$ is a sum of $n+1$ squares of integral linear forms:

$$q_{A_n} = 2 \sum_{i=1}^n x_i^2 - 2 \sum_{i=1}^{n-1} x_i x_{i+1} = x_1^2 + (x_1-x_2)^2 + \dots (x_{n-1}-x_n)^2 + x_n^2$$

It is equally easy to see that $q_{D_n}$ is a sum of $n$ squares. It is a little harder to see but I think is true that $q_{E_n}$ ($n=6,7,8$) is not a sum of $\ge n$ squares of integral forms.

Question: is this a standard fact, well-known to experts? Is there a standard reference? (I hate to reinvent a bycicle.) And has this fact been used for something interesting? (I have an interesting application in mind, so I am looking for connections...)

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 In fact, I think $E_n$ is not a sum of ANY number of squares of integral linear forms. – VA Sep 30 2010 at 16:05

If a quadratic form $x^TAx$ (for lattice $L$) in $n$ variables is expressible as a sum of $m (\ge n)$ squares of integral linear forms
then $x^TAx=||Fx||^2$, where $F$ is an $m \times n$ integral matrix and the columns of $F$
gives an explicit embedding of $L$ as a sublattice of $\mathbb{Z}^m$. For $E_8$ and $m=8$, this will mean
$E_8=\mathbb Z^8$ since both have the same volume, which is not possible. Also from $A=F^TF$ and the Cauchy-Binet formula,
$\det A$ is a sum of $m \choose n$ integral squares which are squared volume of the projections.
Since $\det E_8=1$, there can only be a single term 1 and the rest are zero and this means $E_8$ is embedded inside some
$\mathbb Z^8$ inside $\mathbb Z^m$ so this reduces to the case $m=8$. For $E_6,E_7$, the case $m=n$ can be ruled out since $\det A$ is
not a square as Robin Chapman noted. For $m>n$, since $2=1+1,6=1+1+4=1+1+1+1+1+1$ are the only partition of $\det A$ a a sum of squares, many projections have to be zero which means large $m$ can be reduced to smaller $m$. Can this be used to reduce to the case $m=n$ ?

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 The argument shows that an unimodular lattice not a $\ZZ^n$ cannot be embedded in any $\ZZ^m$ ie. its quadratic form cannot be expressed as a sum of squares of any number of linear forms – unknown (google) Jul 18 2011 at 2:27

Fairly recent work by Ellenberg and Venkatesh, later improved by Schulze-Pillot, show that it is reasonable to hope that $n+3$ squares of linear forms suffice. See Theorem 11, bottom of pdf page 8

Of the hypotheses involved, the more serious is that of sufficiently large minimum, as your quadratic forms have very small minima. Well, if you have favorite expressions for the $E_n$ quadratic forms, let me know, I can probably program something definitive up to $n+3.$

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Although the paper of Ellenberg-Venkatesh you quoted is really interesting, I don't see how to easily apply it: it only applies to forms with large minima, plus there are local conditions to check. I actually think it does not apply at all, since neither of $A_n,D_n,E_n$ is a sum of $\ge n+2$ squares. In any case, I can prove the statement easily, so my questions were (1) standard reference or a 2-line proof? and (2) did this simple characterization of $A_n,D_n,E_n$ show up in relation with something interesting? – VA Sep 30 2010 at 20:09
Dear VA, I see what you mean, about $\geq n+2.$ I found nice Gram matrices for $E_6, E_7, E_8$ on Sloane's website, I am beginning to see, by hand, that they may not be sums of squares by pigeonhole arguments, anyway  www2.research.att.com/~njas/lattices/E6.html  www2.research.att.com/~njas/lattices/E7.html  www2.research.att.com/~njas/lattices/E8.b.html  – Will Jagy Sep 30 2010 at 20:38

If a quadratic form in $n$ variables is the sum of the squares of $n$ integer linear forms, it's the sum of the squares of $n$ rational linear forms. Thus it's equivalent as a rational quadratic form to $x_1^2+\cdots+x_n^2$. In particular its discriminant is a square. This rules out $E_6$ and $E_7$.

The case of $E_8$ is trickier, as it is equivalent over the rationals to $x_1^2+\cdots+x_8^2$. This time you have to show there's no equivalence over the integers, but one form takes solely even values and the other doesn't.

Added The first time round I didn't clock the $\ge n$ condition. But there's certainly a way to put an upper bound on the number $m$ of linear forms one needs.

I'll stick to the $E_8$ form. One can think of this as describing a lattice $L$ in Euclidean space. This lattice $L$ is self-dual, unimodular and even. Its shortest vectors form the set of 240 roots $R$. This set of roots has the nice property (I think it may be called the eutactic property or the perfect property; all this is in Martinet's book on lattices) that $x\mapsto\sum_{y\in R}(x\cdot y)^2$ is proportional to the quadratic form $x\mapsto x\cdot x$. I think actually $\sum_{y\in R}(x\cdot y)^2=30x\cdot x$ as $30=240/8$. Now an integer linear form is a linear form taking integer values on the lattice $L$ and so is $x\mapsto x\cdot z$ for some $z$ in the dual of $L$, so here $z\in L$. If $x\cdot x=\sum_{j=1}^m(x\cdot z_j)^2$ for $z_j\in L$ then $$480=2|R|=\sum_{y\in R}y\cdot y=\sum_{j=1}^m\sum_{y\in R}(y\cdot z_j)^2 =30\sum_{j=1}^m z_j\cdot z_j.$$ Each $z_j\cdot z_j\ge 2$ so we must have $m=8$ and each $z_j\cdot z_j=2$.

You should check my numbers... For $E_6/E_7$ the dual lattice is different from the original but I'm sure they still have the eutactic(?) property. Any way one can get an effective upper bound on $m$, probably not much bigger than $n$.

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This does not rule out that $E_n$ is a sum of $>n$ squares. But it's a good observation. – VA Sep 30 2010 at 16:09
To amplify VA's remark, this argument "rules out" most of $A_n$ as well, but of course the corresponding quadratic forms can be represented as sums of $n+1$ squares, per the original question. – Victor Protsak Sep 30 2010 at 16:39
For simply laced root systems the "eutactic" property you need is the corollary to Theorem 1 of paragraph 6 of chapter 5 of Bourbaki's "Lie Groups and Lie Algebras". The 30 that appears is, of course, the Coxeter number of $E_8$. – Sheikraisinrollbank Oct 1 2010 at 9:12
Robin, thank you for your answer. I think I can prove all my statements quite easily by a direct combinatorial argument, working with the Dynkin graph. But you give a nice argument. – VA Oct 1 2010 at 15:22