Symmetry in semidefinite programs

TL;DR

This paper presents a systematic method using representation theory to simplify symmetric semidefinite programs, illustrated through block diagonalization of the Terwilliger algebra in the binary Hamming scheme.

Contribution

It introduces a general explicit procedure for simplifying symmetric semidefinite programs based on finite group representation theory.

Findings

Derived block diagonalization of the Terwilliger algebra

Connected the algebra to orthogonal Hahn and Krawtchouk polynomials

Provided a tutorial framework for symmetry exploitation in SDPs

Abstract

This paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group. The procedure is based on basic notions of representation theory of finite groups. As an example we derive the block diagonalization of the Terwilliger algebra of the binary Hamming scheme in this framework. Here its connection to the orthogonal Hahn and Krawtchouk polynomials becomes visible.