Writing projective representations over subfields
Let G = 〈 X 〉 be an absolutely irreducible subgroup of GL ( d , K ) , and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL ( d , F ) . K × , where K × denotes the centre of GL ( d , K ) . If...
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| Published in | Journal of algebra Vol. 295; no. 1; pp. 51 - 61 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
2006
|
| Online Access | Get full text |
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| Summary: | Let
G
=
〈
X
〉
be an absolutely irreducible subgroup of
GL
(
d
,
K
)
, and let
F be a proper subfield of the finite field
K. We present a practical algorithm to decide constructively whether or not
G is conjugate to a subgroup of
GL
(
d
,
F
)
.
K
×
, where
K
×
denotes the centre of
GL
(
d
,
K
)
. If the derived group of
G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs
O
(
|
X
|
d
3
+
d
2
log
|
F
|
)
arithmetic operations in
K. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of
GL
(
d
,
K
)
. |
|---|---|
| ISSN: | 0021-8693 1090-266X |
| DOI: | 10.1016/j.jalgebra.2005.03.037 |