Approximate Representations and Approximate Homomorphisms
Cristopher Moore, Alexander Russell

TL;DR
This paper investigates approximate representations of finite groups, establishing bounds on their quality based on the ratio of the representation dimension to the smallest nontrivial representation, with implications for approximate homomorphisms and quasirandom groups.
Contribution
It provides new bounds on approximate representations and homomorphisms of finite groups, linking quasirandomness to embedding limitations and demonstrating the tightness of these bounds.
Findings
Bounds relate approximation quality to the ratio d/d_min
Approximate homomorphisms are limited when target groups have smaller representations
Bounds are shown to be tight through analysis of genuine representations
Abstract
Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. We interpret these…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Finite Group Theory Research · Complexity and Algorithms in Graphs
