TY - JOUR
T1 - Irreducibility of tensor squares, symmetric squares and alternating squares
AU - Magaard, Kay
AU - Malle, Gunter
AU - Tiep, Pham Huu
PY - 2002
Y1 - 2002
N2 - We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteristic, either V is a Weil representation of a symplectic or unitary group, or G is one of a finite number of exceptions. For G in even characteristic, we derive upper bounds for the dimension of V which are close to the minimal possible dimension of nontrivial irreducible representations. Our results are complete in the case of complex representations. We will also answer a question of B. H. Gross about finite subgroups of complex Lie groups script G sign that act irreducibly on all fundamental representations of script G sign.
AB - We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteristic, either V is a Weil representation of a symplectic or unitary group, or G is one of a finite number of exceptions. For G in even characteristic, we derive upper bounds for the dimension of V which are close to the minimal possible dimension of nontrivial irreducible representations. Our results are complete in the case of complex representations. We will also answer a question of B. H. Gross about finite subgroups of complex Lie groups script G sign that act irreducibly on all fundamental representations of script G sign.
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U2 - 10.2140/pjm.2002.202.379
DO - 10.2140/pjm.2002.202.379
M3 - Article
AN - SCOPUS:0036408236
SN - 0030-8730
VL - 202
SP - 379
EP - 427
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -