# A combinatorial question about orthonormal bases

Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. Does it follow that $F$ is a constant function?

This is clearly false for $n=2$. I am wondering if it is true for sufficiently large $n$.

ADDED LATER The $\mathbb R$-valued examples in Cranch's answer may be combined into a single example $v\mapsto v\otimes v$ with values in $\mathbb R^n\otimes \mathbb R^n$, or $n\times n$ matrices. Its image generates the group of symmetric matrices with integer trace. It seems reasonable to expect that every continuous real-valued example comes from this one -- in other words has the form $v\mapsto B(v,v)$ for symmetric bilinear $B$. Maybe this can be worked out using Sawin's suggestion about representations of $O(n)$. But I was also curious about the general case, where the target group might not be (uniquely) divisible.

• The group $G = \{e\}$ is abelian :) – Igor Rivin Apr 26 '15 at 20:24
• Igor, I do not get the joke. – Tom Goodwillie Apr 26 '15 at 21:09
• There was not any, I was confused. – Igor Rivin Apr 26 '15 at 21:58

## 2 Answers

Given a vector $u$ and an orthonormal basis $x_1,\ldots,x_n$, we have $||u||^2 = \left<u,x_1\right>^2 + \cdots + \left<u,x_n\right>^2$. But that means that, if you choose a nonzero $u$, then the function $F(x) = \left<u,x\right>^2$ gives a counterexample.

• I sometimes wish MathOverflow was the kind of forum where one could just say "Great question! Ask your best undergraduates: they'll enjoy telling you how to do it more than I would." – James Cranch Apr 26 '15 at 21:03
• James, I hope you enjoyed it. It's such an attractive and edifying answer, too. – Tom Goodwillie Apr 26 '15 at 21:09
• Do these and constants generate all such functions? – Douglas Zare Apr 26 '15 at 21:13
• With the axiom of choice, there are lots of discontinuous automorphisms of $\mathbb{R}$. However, it might be feasible to classify the continuous functions $S^{n-1} \to \mathbb{R}$ satisfying the condition. Are they all in the closure of the linear combinations of constants and Cranch's functions? – Douglas Zare Apr 26 '15 at 23:15
• @DouglasZare For any such function, pullback by $O(n)$ gives another such function. So the space of such functions is a representation of $O(n)$, and because continuous functions decompose into spherical harmonics, this will decompose into different irreducible representations of $O(n)$. So you just have to test the different spherical harmonics. – Will Sawin Apr 27 '15 at 1:09

For the group $\mathbb{R}$, this was considered a long time ago by Andrew Gleason to solve a problem in the foundations of quantum mechanics.

http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050

Such maps are called frame functions. For $n \geq 3$, frame functions $S^{n-1} \rightarrow \mathbb{R}$, taking positive values correspond to positive-definite operators on $\mathbb{C}^n$. When the sum is chosen to be 1, one gets density matrices.

Of course, if one drops the positivity and boundedness requirements, one can get horribly complicated functions using a nonlinear group automorphism of $\mathbb{R}$.

I am interested as to Tom Goodwillie's reasons for considering this problem.