B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. 3 (1990), 929 \Gamma 960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of type H = S Delta 2 of S with Z(H) Z(S) CH (S) which are isoclinic to each other. Therefore, one should always specify, which double extension of S we are working with. For more details on isoclinic groups cf. Lemma 2.11 [29] The same lattices have been considered by B. H. Gross [15] in the context of the so called globally irreducible representations (for the precise definition see [15] 29] When p j 3 mod 4 and (1) p Gamma 1) 2, the representation V is globally irreducible. Gross also shows that, for p odd, there are two globally irreducible representations V of ....

....one should always specify, which double extension of S we are working with. For more details on isoclinic groups cf. Lemma 2.11 [29] The same lattices have been considered by B. H. Gross [15] in the context of the so called globally irreducible representations (for the precise definition see [15], 29] When p j 3 mod 4 and (1) p Gamma 1) 2, the representation V is globally irreducible. Gross also shows that, for p odd, there are two globally irreducible representations V of Sp 2n (p ) of dimension p Gamma 1 over Q , which are related to the Weil representations and lead ....

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B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. 3 (1990), 929 \Gamma 960.

....a representation of Sp(2n; O=p l ) for some l. Many of our results hold in the case that R is an arbitrary nite commutative ring of odd characteristic, and we work in this generality at rst. The theory starts o in a similar fashion to the case where R is a nite eld (see [G er77] Isa73] [Gro90], Sze98] We start with a nitely generated R module V which has a non degenerate alternating bilinear form h ; i with values in R. The symplectic group Sp = Sp(V ) is the group of invertible R endomorphisms of V which preserve the form. We consider the analogue H of the Heisenberg group, H = ....

B. H. Gross, Group representations and lattices, J. Amer. Math. Soc 3 (1990), 929-960.

....m =1, 2, 3, andpr#5k t aconjectur#j for# ula of the kissing number of L mfor gener#j m. After the fir# ver#Kjk of this paper was finished, theauthor wasinfor#O that Dummigan and Tiep [3] have alsoconsider# the latticesN m prim (X)fr#I agr#BMB theor#k6 point of view using an idea ofGr#IN [8]. Acknowledgment. Par# of this wor# was donedur#M theauthor#O stay at the Max Planck Institute fur Mathematik in Bonn fr#n June to July in 1998. The author would like to thank the Max Planck Institutefor giving him a stimulating r#ating h envir#kjj t. He also would like to thankPr#kMkkIO Eiichi ....

B. H. Gross, `Group representations and lattices', J. Amer. Math. Soc. 3 (1990), 929--960.

....Corollary 3.6 yields [c( Q1;5 ] Let q be a power of an odd prime p. As noted by a referee, the group U 3 (q) has a absolutely irreducible rational representation V of degree q 2 q 1 (cf. Simpson and Frame 1973] and a rational character of degree q(q 1) with Schur index 2 at 1 and p ([Gross 1990, Section 14] So one might hope to generalize this calculation to arbitrary prime powers q. For q = 7 the character occurs with odd multiplicity in W , so here [c( Q1;7 ] If q = 3 or q = 11, then dim(V ) 1 (mod 8) Therefore R splits c( for any positive de nite quadratic form ....

B.H. Gross, Group representations and lattices. J. Amer. Math. Soc. 3 (1990) 929-960.

.... in G(k) for some number field k, or F to be irreducible, when one thinks of G(k) as a subgroup of GL n (k) There are also such concepts which make use of a more arithmetic situation, like Thompson s concept of utter irreducibility (cf. Tho 76] or its generalization to global irreducibility ( Gro 90] These concepts take the finite group F as a primary object. Another approach is suggested by Gross concepts of ZZ models (cf. Gro 96] which take the algebraic group G and certain of its integral forms G as the primary objects to obtain finite groups as G(ZZ) which one might hope are big, ....

B.H. Gross, Group representations and lattices. J. AMS 3 (1990) 929-960.

....by computer. In particular a new extremal even unimodular lattice of rank 48 is constructed. Introduction. The study of finite integral matrix groups is one source for producing nice lattices. In particular representations of the group PSL 2 (p) for p j 3 (mod 4) have been studied in [10] in connection with globally irreducible representations. Gross gives an interpretation of some of the invariant lattices as Mordell Weil lattices. Since the real Schur index of the faithful rational representations of SL 2 (p) of degree 2(p Gamma 1) is two, they can be viewed as representations ....

B.H. Gross, Group representations and lattices, J. AMS 3 (1990), 929-960.

....22 :2, a maximal finite subgroup of GL 80 (Q) isomorphic to a central product of 2:Alt 7 and 2:M 22 :2. The G invariant lattices in Q 80 are of the form cL 80 , where 0 6= c 2 C Q 80 Theta80 (G) Q [ff] is an invertible element in the commuting algebra of G (i.e. G is a GIR in the sense of [Gro 90] Proof: Let U : 2:Alt 7 Omega p Gamma7 2:M 22 :2. By construction U is a subgroup of Aut(L 80 ) G. The commuting algebra of U is isomorphic to Q [ff] and U fixes up to isomorphism only one lattice (cf. CCNPW 85] and [JLPW 95] So we only have to show that G = U . Since L 80 is ....

B. Gross, Group representations and Lattices. J. AMS 3 (1990), 929-960

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B. H. Gross, Group representations and lattices, J. Amer. Math. Soc. 3 (1990), 929 \Gamma 960.

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