The Allcock ball quotient

Abstract

In this article we provide further evidence for the monstrous proposal of Daniel Allcock, by giving a plausible but still conjectural explanation for the deflation relation in the Coxeter group quotient of the orbifold fundamental group.

Two Lorentzian Lattices

“…Using our compact notation the 18 cases can be conveniently summarized in the following table and the lemma follows from the observation that in each row the most right symbol represents a point in the chamber D for Γ 13 The conclusion is that the intersection of the complex mirror arrangement in C ⊗ R V for all norm three roots in L with the real form V consists of the mirror arrangement for all norm one roots in L r together with the transform of this arrangement under the orthogonal involution of V coming from a projective duality on P 2 (3). Hence the results of the previous section indeed prove Conjecture 1.7 made by one of us in [14]. Using the results of that paper we arrive at Theorem 5.2.…”

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We prove a conjecture formulated in [14], which in turn provides a good deal of evidence for the monstrous proposal of Daniel Allcock [2].

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“…The proof of this result is a rather straightforward calculation given in the next section. The interest of this theorem is its analogy with the theorem below, which was conjectured in [14] and provides a positive step towards the monstrous proposal of Daniel Allcock [2]. Now, let us consider the projective plane P 2 (3) over a field of 3 elements, and denote as before by P and L the sets of its 13 points and 13 lines respectively.…”

“…Again the proof is by straightforward but more extensive (however not unpleasant) calculations. The inclusion P ⊂ G was conjectured in [14] and in the final section of this paper we discuss its relevance towards the monstrous proposal of Daniel Allcock [2].…”