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Robert L. Griess Jr

Publications and source records attributed to Robert L. Griess Jr.

11 recordsLinked to original sources
arXiv2014

Lattice-integrality of certain group-invariant integral forms in vertex operator algebras

Certain vertex operator algebras have integral forms (integral spans of bases which are closed under the countable set of products). It is unclear when they (or integral multiples of them) are integral as lattices under the natural bilinear form on the VOA. We show that lattice-integrality may be arranged under some hypotheses, including cases of integral forms invariant by finite groups. In particular, there exists a lattice-integral Monster-invariant integral form in the Moonshine VOA.

math.GR↗
arXiv2009

Midwest cousins of Barnes-Wall lattices

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks $2^{d-1}\pm 2^{d-k-1}$, for odd integers $d\ge 3$ and integers $k=1,2, ..., \frac {d-1}2$. Their minimum norms are moderately high: $2^{\lfloor \frac d2 \rfloor -1}$.

math.NT↗
arXiv2009

Rank 72 high minimum norm lattices

Given a polarization of an even unimodular lattice and integer $k\ge 1$, we define a family of unimodular lattices $L(M,N,k)$. Of special interest are certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a $L(M,N,3)$. We note a few $L(M,N,k)$ in dimensions 96, 120 and 128 with moderately high minimum norms.

math.NT↗
arXiv2005

Involutions on the the Barnes-Wall lattices and their fixed point sublattices, I

We study the sublattices of the rank $2^d$ Barnes-Wall lattices $\bw d$ which occur as fixed points of involutions. They have ranks $2^{d-1}$ (for dirty involutions) or $2^{d-1}\pm 2^{k-1}$ (for clean involutions), where $k$, the defect, is an integer at most $\frac d 2$. We discuss the involutions on $\bw d$ and determine the isometry groups of the fixed point sublattices for all involutions of defect 1. Transitivity results for the Bolt-Room-Wall group on isometry types of sublattices extend those in \cite{bwy}. Along the way, we classify the orbits of $AGL(d,2)$ on the Reed-Muller codes $RM(2,d)$ and describe {\it cubi sequences} for short codewords, which give them as Boolean sums of codimension 2 affine subspaces.

math.GR↗
arXiv2004

The rank two lattice type vertex operator algebras V_L^+ and their automorphism groups

Let L be a positive definite even lattice and V_L^+ be the fixed points of the lattice VOA V_L associated to L under an automorphism of V_L lifting the -1 isometry of L. For any positive rank, the full automotphism group of V_L^+ is determined if L does not have vectors of norm 2 or 4. For any L of rank 2, a set of generators and the full automorphism group of V_L^+ are determined.

math.QA↗
arXiv2004

Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices

We give a new existence proof for the rank 2^d even lattices usually called the Barnes-Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs are relatively free of calculations, matrix work and counting, due to the uniqueness viewpoint. We deduce the labeling of coordinates on which earlier constructions depend. Extending these ideas, we construct in dimensions 2^d, for d>>0, the Ypsilanti lattices, which are families of indecomposable even unimodular lattices which resemble the Barnes-Wall lattices. The number Upsilon (2^d) of isometry types here is large: log_2 (Upsilon (2^d)) has dominant term at least (r/4) d 2^{2d}, for any r in [0, 1/2). The Ypsilanti lattices may be the first explicitly given families whose sizes are asymptotically comparable to the Siegel mass formula estimate (log_2(mass(n)) has dominant term (1/4) log_2(n) n^2). This work continues our general uniqueness program for lattices, begun in Pieces of Eight. See also our new uniquness proof for the E_8-lattice.

math.GR↗