Four-dimensional determinant-one representations and cyclic singlets
Abstract
No finite group has a four-dimensional complex representation with trivial determinant and no trivial singlet whose restriction to every cyclic subgroup contains the trivial singlet. This proves the finite-group conjecture of S. Groot Nibbelink, O. Loukas, A. Mutter, E. Parr and P. K. S. Vaudrevange (arXiv:1710.09237), which the authors checked by computer for the 1,594 finite groups arising from the CARAT classification and for all finite groups of order at most 500; the two collections overlap.
Definition 1.1 (The finite group conjecture).
Formalization. D5/S3/Quantum/Algebra/FourDimensionalSingletBranching.claim (✓ std3).
Citation. Stefan Groot Nibbelink; Orestis Loukas; Andreas Mütter; Erik Parr; Patrick K. S. Vaudrevange (2020). Tension between a vanishing cosmological constant and non-supersymmetric heterotic orbifolds. DOI: 10.1002/prop.202000044. URL: https://arxiv.org/abs/1710.09237v1.
Commentary.
For every finite group H and every representation D of H by 4 x 4 complex matrices (a monoid homomorphism from H to the complex 4 x 4 matrices): if (i) every D(h) has determinant 1 and (ii) the only vector fixed by every D(h) is 0 (D contains no trivial singlet), then (iii) fails, that is, some cyclic subgroup K of H fixes no nonzero vector (the branching of D to K contains no trivial singlet).
Theorem 1.2 (Proof of the conjecture).
Proof. Machine-checked in Lean as D5/S3/Quantum/Algebra/FourDimensionalSingletBranching.result (✓ std3). ∎
Resolves. Problems/groot-nibbelink-2020-four-dimensional-singlet-branching (proved) by D5/S3/Quantum/Algebra/FourDimensionalSingletBranching.result.
Source. Repository-derived.
Acknowledgement. Stefan Groot Nibbelink; Orestis Loukas; Andreas Mütter; Erik Parr; Patrick K. S. Vaudrevange (2020). Tension between a vanishing cosmological constant and non-supersymmetric heterotic orbifolds. DOI: 10.1002/prop.202000044. URL: https://arxiv.org/abs/1710.09237v1.
Commentary.
Suppose (i), (ii) and (iii). For the cyclic subgroup generated by h, (iii) gives a nonzero vector fixed by D(h), so det(1 - D(h)) = 0 for every h. For a 4 x 4 matrix A of determinant 1 the adjugate is the inverse, and expanding the determinant gives det(1 - A) = 2 - tr A - tr A^(-1) + e_2(A) with e_2(A) = ((tr A)^2 - tr(A^2))/2. The sum S of D(h) over H satisfies D(g) S = S, so its columns are fixed vectors and S = 0 by (ii); hence the traces of D(h) and of D(h)^(-1) = D(h^(-1)) sum to 0. Let T be the sum of the Kronecker squares D(h) x D(h) and W the swap of the two tensor factors. Then T^2 = |H| T, W^2 = 1 and W commutes with T, so Q = T(1 - W)/(2|H|) is idempotent; its trace is the rank of Q, a natural number r, and it equals the average of e_2(D(h)) because tr((A x A) W) = tr(A^2). Summing, the sum of det(1 - D(h)) over H is |H|(2 + r), which is not 0, a contradiction.