Group Representations and Lattices

Abstract

This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Sh I,Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irreducibility hypotheses on a rational representation V of a finite group G, then the G-stable Euclidean lattices in V are severely restricted. Unfortunately, these hypotheses are rarely satisfied when V is absolutely irreducible over Q. But there are many examples where the ring End G ( V) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the Mordell-Weillattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves.In § I we discuss lattices and Hermitian forms on T/, and in § §2-4 the strong irreducibility hypotheses we wish to make. In § 5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean I. LATTICES AND HERMITIAN FORMSIn this section, we establish the notation that will be used throughout the paper. Let G be a finite group of order g. Elements of G will be denoted s, t , .... Let V be a finite-dimensional rational vector space that affords a linear representation of Gover Q. We view elements of G as linear operators acting on the right of V, and so have the formula: v S / = (v s / for v E V and s, t E G. Let x(s) = Tracev(s) be the character of V.

On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings

“…The equality N (A (r) p−1 ) = 2r holds for r = 1, r = 2, r = 3 and also for r = (p + 1)/4 with p ≡ 3 mod 4. This last case was proved by Elkies (cited in Gross [Gross 1990]), from the general theory of Mordell-Weil lattices developped by Elkies and Shioda concerning the groups of rational points of elliptic curves over function fields [Shioda 1992]. The equality N (A (r) p−1 ) = 2r was also proved to be true for p ≤ 37 and r ∈ [1, Using the assertion (ii) in Theorem 5.3 we obtain the following Proposition.…”

Weil representations of finite general linear groups and finite special linear groups

“…(i) First we consider the case`D 0, or more generally, ' lifts to a complex character of G. Using the character table of G given, e.g., in [Geck et al 1996], one can check that the relation 'OE1 D 0 implies that ' is a Weil character. (Note that the character table of G was first determined in [Steinberg 1951].…”

“…Similarly, let 1 , 2 , and 3 denote the Brauer characters of the irreducible ‫ކ‬G-modules D.1; / ı D.b; .1// " G, with D .3/, .2; 1/, and .1 3 /, respectively. Using [Geck et al 1996], we can compute…”

“…Weil representations of finite symplectic groups Sp 2n .q/ with 2 j q were constructed by R. M. Guralnick and the author in [Guralnick and Tiep 2004]. Weil representations attract much attention because of their many interesting features; see, for instance, [Dummigan 1996;Dummigan and Tiep 1999;Gow 1989;Gross 1990;Scharlau and Tiep 1997;1999;Tiep 1997a;1997b, Zalesski 1988.…”

“…The construction of the Weil representations may be found in [Howe 1973;Gérardin 1977;Guralnick and Tiep 2004;Seitz 1975], etc. In particular, in the case of general and special linear groups, they can be constructed as follows.…”