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Fulton W., Harris J. Representation theory. A first course (Springer, 1991)(L)(T)(ISBN 0387974954)(285s).djvu |
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a problem; there are many other ways of describing these isomorphisms, and readers
who disagree with our choice can substitute their own...
The Lie algebra so2n+1 C
is correspondingly the space of matrices X satisfying the relation 'X ¦ M +
M ¦ X = 0; if we write X in block form as
0
/...
To start, to choose an ordering of the roots
we take as linear functional on I)* a form / = qi^ + ••• + cnHn, where
c, > c2 > • • • > cn > 0...
To see in this case that the map is an
isomorphism, consider the tensor product V — U ® W of the pullbacks to
sl2C x sI2C of the standard representations of the two factors...
Thus, there is no
way of constructing all the representations of so4C by applying linear- or
multilinear-algebraic constructions to the standard representation; it is only
after we are aware of the isomorphism so4C ? sI2C x sl2C that we can
construct, for example, the representation r(Li+z.2)/2 with highest weight
(L[ + L2)/2 (of course, this is just the pullback from the first factor of
sI2C x sI2C of the standard representation of sl2C)...
Is it equal to the intersection of these
kernels? Show that the weight diagram of this representation is
After you are done with this analysis, compare with the analysis given of the
corresponding representation in Lecture 16...
In order to generate all the representations,
we still need to find the irreducibie representations with highest weight along
the remaining two edges of the Weyl chamber...
In fact (as the statement of the theorem implies), two of them
are: It is not hard to check that, in fact, w(n>O) and w'"'1' are killed by every
positive root space gt(+Lj...
The square of this map is the identity, and
decomposing A"V into +1 and — 1 eigenspaces for this map gives two
subrepresentations...
Representations of the Even Orthogonal Algebras
291
y = alLl +¦¦¦ + an_2(L, + ••• + Ln_2)
+ an_1(L1 + ¦ ¦ ¦ + Ln_, - LJ/2 + a,(U +¦¦¦ + LJ/2
with a, g M...
We can rule out this
possibility by direct calculation: for example, if this were the case, then A3K
would contain a highest weight vector with weight Lt...
Moreover, as we saw in the cases of so5C and so7C, the exterior powers of
the standard representation do serve to generate all the irreducible representa-
representations whose highest weights are in the sublattice Z{L1;..., ?„}: in general we
have the following theorem...
Representations of associated partitions restrict to isomorphic representa-
representations of SOmC...
This covering is even easier to see for the entire orthogonal group O3U,
which is generated by reflections Rv in unit vectors v (with + v determining
the same reflection): we can describe the double cover of O3IR as the group
generated by unit vectors v, with relations
«V ¦¦••"« = "'i ¦ • • •' wm
whenever the compositions of the corresponding reflections are equal, i.e.,
whenever
and also relations
(— v) ¦ (— w) = v ¦ w
for all pairs of unit vectors v and w...
This in turn determines either one or two representations of
the orthogonal Lie algebras, which turn out to be the representations which
were needed to complete the story in the last lecture...
C\ C+-C~<= C~, C~C+ c C, C~-C~ cC(; C+ is spanned
by products of an even number of elements in Kand C~ is spanned by products
of an odd number...
(b) Deduce that every element of the orthogonal group of Q can be written
as the product of at most 2dim(F) reflections...
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