Group representations that resist worst-case sampling

TL;DR

This paper constructs groups with irreducible representations that maintain high operator norm over small subsets, answering longstanding questions about group expansion and sampling resistance.

Contribution

It demonstrates the existence of infinite groups with representations resistant to worst-case sampling, using affine groups over finite fields and novel almost-invariant set constructions.

Findings

Existence of groups with representations resistant to small-sample averaging

Construction based on affine groups over finite fields

Sets that are almost invariant under additive and multiplicative translations

Abstract

Motivated by expansion in Cayley graphs, we show that there exist infinitely many groups $G$ with a nontrivial irreducible unitary representation whose average over every set of $o(\log\log|G|)$ elements of $G$ has operator norm $1 - o(1)$. This answers a question of Lovett, Moore, and Russell, and strengthens their negative answer to a question of Wigderson. The construction is the affine group of $\mathbb{F}_p$ and uses the fact that for every $A \subset \mathbb{F}_p\setminus\{0\}$, there is a set of size $\exp(\exp(O(|A|)))$ that is almost invariant under both additive and multiplicatpive translations by elements of $A$.

Equations

$$\biggl{\|}\frac{1}{|S|}\sum_{g\in S}\rho(g)\biggr{\|}_{\mathrm{op}}\leq 1-\epsilon.$$

$${\mathbb{P}}_{g_1,\dots ,g_k\in G}\left[{\Vert \frac{{\rho }^n(g_1)+\cdots +{\rho }^n(g_k)}{k}\Vert }_{\mathrm{op}}=1\right]=1-o(1).$$

$${\Vert \frac{\rho (g_1)+\cdots +\rho (g_k)}{k}\Vert }_{\mathrm{op}}\ge 1-ϵ$$

$$|a\cdot X\setminus X|\le ϵ|X|\text{and}|(a+X)\setminus X|\le ϵ|X|\text{for all }a\in A.$$

$$P$$

$$Q$$

$$X=Q^{-1}\biggl{(}\sum_{y\in Q}y\cdot P\biggr{)},$$

$$y_{0}^{-1}\biggl{(}\sum_{y\in Q}yx_{y}\biggr{)}$$

Full text

The construction is the affine group of $\mathbb{F}_{p}$ and uses the fact that for every $A\subset\mathbb{F}_{p}\setminus\{0\}$, there is a set of size $\exp(\exp(O(|A|)))$ that is almost invariant under both additive and multiplicatpive translations by elements of $A$.

The author was supported by an Esmée Fairbairn Junior Research Fellowship at New College, Oxford and a Research Fellowship at the Simons Institute, Berkeley.

Let $G$ be a finite group and $\rho$ a unitary representation of $G$. For a subset $S\subset G$, we say that $S$ is $\epsilon$-expanding with respect to $\rho$ if

[TABLE]

Otherwise, we say that $\rho$ $\epsilon$-resists $S$. We say that $S\subset G$ is $\epsilon$-expanding if it is $\epsilon$-expanding with respect to every non-trivial irreducible unitary representation of $G$, which is essentially the same as saying that the adjacency matrix of the Cayley graph on $G$ generated by $S$ has all eigenvalues, except the top one, bounded by $1-\epsilon$ in absolute value. It is closely related to a more combinatorial notion of expansion in graphs via Cheeger’s inequality.

By a theorem of Alon and Roichman [1] on eigenvalues of random Cayley graphs, for any group $G$, a random set of $C_{\epsilon}\log|G|$ group elements is $\epsilon$-expanding with high probability. This bound is tight for abelian groups, up to constant factors. For example, when $G=(\mathbb{Z}/2\mathbb{Z})^{n}$, it takes $n=\log_{2}|G|$ elements simply to generate the group.

On the other hand, for certain families of “highly non-abelian” groups, including all non-abelian simple groups, a bounded number of generators suffices to obtain $\epsilon$-expansion. In certain cases, such as $\operatorname{SL}_{2}(\mathbb{F}_{p})$ [2], and more generally, any finite simple groups of Lie type of bounded rank [3], we know that $\{g,h,g^{-1},h^{-1}\}$ is $\epsilon$-expanding with high probability for uniformly random group elements $g$ and $h$. See surveys [4, 6] for more on expansion.

Wigderson conjectured [8] in his 2010 Barbados lectures that there is some constant $k$ so that for any finite group $G$ and a nontrivial irreducible unitary representation $\rho$, a list of $k$ random elements of $G$ is $(1/2)$-expanding with respect to $\rho$ with probability at least $1/2$. Note that this is true for abelian groups, where every irreducible representation is one-dimensional, even though it takes $C\log|G|$ elements to expand with respect to every non-trivial irreducible representation simultaneously.

Wigderson’s conjecture was disproved by Lovett, Moore, and Russell [5], who found an infinite family of groups such that, with high probability, a random subset of $k$ elements does not expand at all with respect to a specific nontrivial irreducible representation. More specifically, they showed that if $K$ is a fixed non-abelian group with trivial center (e.g., $K=S_{3}$), and $\rho$ is a faithful irreducible unitary representation of $\rho$, then, $G=K^{n}$ with the irreducible representation $\rho^{n}=\rho\otimes\cdots\otimes\rho$ has the property that, as $n\to\infty$, provided that $k=o(\log n)$, one has

[TABLE]

Therefore, there are infinitely many groups $G$ with a non-trivial irreducible unitary representation that resist any set of size $o(\log\log|G|)$. Despite these negative results for random group elements, they asked whether there are constants $k$ and $\epsilon$ such that for any group $G$ and any nontrivial irreducible representation $\rho$, there exist some $k$ elements of $G$ that $\epsilon$-expand respect to $\rho$. We answer this question in the negative.

[TABLE]

for any $g_{1},\dots,g_{k}\in G$ with $k\leq c_{\epsilon}\log\log|G|$.

More succinctly, there exist groups $G$ with a representation that $o(1)$-resists any set of $o(\log\log|G|)$ elements (the construction in [5] works for a random set, whereas ours works for all sets).

This gives a strong negative answer to Wigderson’s question, as it shows that there no choice of a constant number of elements of $G$ can $\epsilon$-expand with respect to $\rho$, let alone a random choice.

We prove Theorem 1 by taking $G=\operatorname{Aff}(\mathbb{F}_{p})$, the affine group of $\mathbb{F}_{p}$. Its elements are affine transformations $x\mapsto ax+b$, where $a\in\mathbb{F}_{p}^{\times}$ and $b\in\mathbb{F}_{p}$. Let $\rho$ denote its standard representation with the trivial component removed. Theorem 1 for $\operatorname{Aff}(\mathbb{F}_{p})$ is an immediate consequence of the following result.

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Here we adopt the standard notation from additive combinatorics: $a\cdot X:=\{ax:x\in X\}$ and $a+X:=\{a+x:x\in X\}$. Theorem 1 for $G=\operatorname{Aff}(\mathbb{F}_{p})$ follows as a corollary of Theorem 2. Indeed, if $S$ consists of affine maps $x\mapsto a_{i}x+b_{i}$, then take $A$ to be the set of all nonzero elements that appears as $a_{i}$ or $b_{i}$ for some $i$. The claim (1) follows by considering the characteristic vector of $X$, appropriately normalized (noting $|X|\leq p/2$ if $|S|\leq c_{\epsilon}\log\log p$; we may need to rescale $\epsilon$ by a constant factor).

A proof of Theorem 2 was given by Terry Tao in a MathOverflow post [7].111The author thanks Ben Green for pointing out [7] to him. We include the proof here for completeness.

[TABLE]

Let

[TABLE]

i.e., the set of all elements that can be written as

[TABLE]

for some choices of $y_{0}\in Q$ and $x_{y}\in P$ for each $y\in Q$. It is easy to check (2), as $|(a+P)\setminus P|\leq\epsilon|P|$ and $|(a\cdot Q)\setminus Q|\leq\epsilon|Q|$ for any $a\in A$. We have $|X|\leq|Q||P|^{|Q|}\leq L^{k}L^{kL^{k}}\leq e^{(1/\epsilon)^{O(k)}}$. ∎

It remains an open question whether the bounds in Theorems 1 and 2 can be improved. We conjecture that they cannot.

Note that Alon–Roichman theorem implies that Conjecture 3 is true if we replace $\log\log|G|$ by $\log|G|$ (by taking a random $S$). We do not know any further improvements.

Acknowledgment

The author thanks Ben Green for discussion and Shachar Lovett for encouraging him to write up this result.