Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

The Reflection-Twisted Lattice Ground Representation

Abstract

The actual reflection quotient of an even unimodular lattice has an irreducible complex ground representation.

Let D have finite rank r, an integral symmetric Gram matrix G, and even diagonal. Charges are L=(Fin(r) to Z). Write B(a,b)=sum_i,j a_i G_ij b_j and let c(a,b) be the lower-triangular cocycle exponent, including G_ii/2 on the diagonal, used by the actual lattice fields. The sign extension E consists of pairs (s,a), where s is in Z/2 and zero denotes the positive sign. Its product is (s,a)(t,b)=(s+t+c(a,b),a+b), with the first coordinate reduced modulo two. The complex sign character sends s to 1 for s=0 and to minus one for s=1, and is an injective homomorphism into the complex units.

Charge reflection lifts to theta(s,a)=(s,-a). Define K to be the subgroup generated by all g inverse times theta(g). Each such generator is (0,-2a). Doubled charges with positive sign are central. Cocycle reduction gives a surjective homomorphism E to the signed cocycle extension of L/2L, whose kernel consists exactly of (0,2a). Both inclusions identify this kernel with the generated K, so E/K is the actual reduced extension. The cocycle identity c(a,b)+c(b,a)=B(a,b) gives c(a,a)=B(a,a)/2 over the integers.

Theorem 1.1 (Irreducibility, central sign, squares and ground dimension).

Lean statement: D5/S3/VertexAlgebra/LatticeTwistedGroundRealization.actual_twisted_ground_representation

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeTwistedGroundRealization.actual_twisted_ground_representation (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

If det(G) is a unit in Z, there exist a natural number n and a complex representation rho of E/K on functions (Fin(n) to Z/2) to C. The rank satisfies r=n+n. The representation is irreducible, and the negative central sign acts as minus the identity. Its complex dimension is 2^(r/2); within this same construction rank 24 gives 2^12. For every integral charge a, the square of rho([0,a]) is (-1)^(B(a,a)/2) times the identity. For all charges a,b, rho([0,a])rho([0,b]) equals (-1)^B(a,b) times rho([0,b])rho([0,a]).

The Gram matrix reduced modulo two is nondegenerate because its determinant remains a unit. Even diagonal and cocycle symmetrization make its pairing alternating, including in characteristic two. A symplectic basis gives coordinates v=(x,y), with x,y in (Z/2)^n. Thus even rank is derived. Let C be the matrix of the actual cocycle in this basis, and let P(v,w)=dot(right(w),left(v)). The matrix S=C+P is symmetric over Z/2.

For each diagonal entry S_ii choose the complex fourth root z_i=1 when S_ii=0 and z_i=i when S_ii=1. In an ordered coordinate set put N(v)=sum_(i<j) S_ij v_i v_j and phi(v)=(product_i z_i^(v_i.val)) (-1)^N(v). The finite carry identity proves phi(v)phi(w)=(-1)^S(v,w) phi(v+w). These phases retain the square of each lattice lift, including lifts that square to the negative sign.

The concrete operator W(x,y) acts by (W(x,y)f)(t)=(-1)^dot(y,t) f(t+x). Its product cocycle is P. The operator assigned to the actual reduced element (s,v) is (-1)^s phi(v) W(v). The carry identity verifies the original cocycle C multiplication, and composition with the proved quotient isomorphism gives rho on E/K.

An invariant complex subspace is stable under every shift and sign diagonal, since the phase of every group element is nonzero. The coordinate projector (1+(-1)^t_i D_i)/2 keeps exactly the coordinate value t_i. Their finite product extracts f(t) times the delta function at t. A nonzero vector therefore yields a delta function; shifts yield every delta function, which form the basis of the whole function space. This proves irreducibility and identifies its matrix coefficients.

Bakalov and Kac, Twisted Modules over Lattice Vertex Algebras, arXiv math/0402315v1, printed page 16, equation (4.53), relates the ground dimension to the finite central quotient; Proposition 4.4 on pages 16-17 describes its irreducible representations, and Remark 4.4 on page 18 separates the ground factor from the twisted oscillators. Their extension uses the complex multiplicative group. The sign-extension comparison and the explicit operators here are proved for the stated lower-triangular cocycle. No comparison with their extension, positivity, rootlessness, Leech classification, twisted VOA construction or Monster action is asserted.

References