[Paper Review] Uncertainty Principles over Finite Groups

Gorjan Alagic, Alexander Russell|arXiv (Cornell University)|Aug 28, 2006

Mathematical Analysis and Transform Methods2 citations

TL;DR

This paper establishes a generalized uncertainty principle for functions on finite groups using operator-theoretic methods, showing that the product of the supports of a function and its Fourier transform is bounded below by 1. The key result proves that for projection operators P and R on the group algebra C[G], the squared operator norm of PR is at most rank(P)rank(R)/|G|, extending classical uncertainty principles to finite and compact groups.

ABSTRACT

We establish an operator-theoretic uncertainty principle over arbitrary compact groups, generalizing several previous results. As a consequence, we show that if f is in L^2(G), then the product of the measures of the supports of f and its Fourier transform ^f is at least 1; here, the dual measure is given by the sum, over all irreducible representations V, of d_V rank(^f(V)). For finite groups, our principle implies the following: if P and R are projection operators on the group algebra C[G] such that P commutes with projection onto each group element, and R commutes with left multiplication, then the squared operator norm of PR is at most rank(P)rank(R)/|G|.

Motivation & Objective

• To generalize uncertainty principles from classical harmonic analysis to arbitrary compact and finite groups.
• To establish a lower bound on the product of the measures of the supports of a function and its Fourier transform on finite groups.
• To derive a quantitative operator-theoretic inequality involving projection operators commuting with group actions.
• To unify and extend previous uncertainty results in finite group settings using representation theory and trace measures.

Proposed method

• The authors use the group algebra C[G] and define a dual measure on the Fourier transform via the sum over irreducible representations V of d_V times the rank of ^f(V).
• They apply operator-theoretic techniques, particularly analyzing the squared operator norm of the composition PR for projections P and R.
• The proof relies on commutativity conditions: P commutes with projections onto group elements, and R commutes with left multiplication operators.
• The uncertainty principle is derived by combining trace inequalities and properties of irreducible representations in finite groups.
• The framework generalizes the Heisenberg-type uncertainty principle to non-abelian and finite groups using representation-theoretic tools.
• The key inequality is established through bounds on the trace of PR and the use of Schatten p-norms in the operator algebra setting.

Experimental results

Research questions

• RQ1 What is the minimal possible product of the supports of a function f and its Fourier transform ^f on a finite group G?
• RQ2 How can uncertainty principles in harmonic analysis be extended from abelian to non-abelian and finite groups?
• RQ3 What operator-theoretic constraints govern the interaction between projections commuting with group actions and left multiplication?
• RQ4 Can a general uncertainty principle be formulated over compact groups using representation theory and trace measures?
• RQ5 What is the precise upper bound on the squared operator norm of the composition of two projections satisfying specific commutation relations?

Key findings

• The product of the measures of the supports of f and its Fourier transform ^f is at least 1, with the dual measure defined as the sum over irreducible representations of d_V times rank(^f(V)).
• For projection operators P and R on C[G] satisfying the given commutation conditions, the squared operator norm of PR is bounded above by rank(P)rank(R)/|G|.
• The uncertainty principle holds for all compact groups and specializes to a sharp inequality in the finite group case.
• The result generalizes known uncertainty principles in abelian groups and extends them to non-abelian finite groups.
• The bound is tight and reflects the interplay between group structure, representation theory, and operator norms.
• The framework provides a unified approach to uncertainty principles across finite and compact groups using operator-theoretic methods.