Symmetric Squares, Spherical Designs, and Lattice Minima

Abstract

Journal of Algebra 240 (2001) 185-208. doi:10.1006/jabr.2000.8713Received by publisher: 2000-05-05Harvest Date: 2016-01-04 12:19:58DOI: 10.1006/jabr.2000.8713Page Range: 185-20

Irreducibility of tensor squares, symmetric squares and alternating squares

“…The irreducible X(V ) for complex Weil modules V of SU n (2) and SU n (3) are determined in [20] and [14].…”

“…In particular X(V ) is reducible unless the complex representation X(V ) is irreducible. The latter cannot happen according to [14], Prop. 3.8.…”

“…The Weil modules of Sp 2n (q), q = 3, 5, 9, and of SU n (q), q = 2, 3, do indeed give irreducible examples. In characteristic = 0, this question has been resolved in [19] for Sp 2n (q), and in [20,14] for SU n (q). …”

“…Gross observed (cf. for instance [14]) that U 3 (3), resp. 3 D 4 (2), give rise to examples for type G 2 , resp.…”

“…(1) V is the heart of the natural permutation module of G = A n (see [18]), (2) V is a Weil module of G = Sp 2n (q), q ∈ {3, 5, 9}, (see [19,20]), V is a Weil module of G = SU n (q), q ∈ {2, 3} (see [20,14]), (4) V is a module of dimension (2 n − 1)(2 n−1 − 1)/3, (2 n + 1)(2 n−1 + 1)/3 or (2 2n − 1)/3 for G = Sp 2n (2) and X = Λ 2 (see Prop. 7.4).…”