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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 bilinear form Q may be expressed as
Q{x,y) = 'x-M-y,
where M is the {In + 1) x Bn + 1) matrix
M =
(the diagonal blocks here having widths n, n, and 1)...
SOmC and somC
271
are just the pairwise distinct sums ±Lf ± Ly In the odd case m - In + 1, we
see that e2n+I e V is an eigenvector for the action of h with eigenvalue 0, so
that the weights of the standard representation V are { + L,} u {0} and the
weights of the adjoint representation correspondingly {±Lt ± L}) u {±Lj...
The positive roots in the case of so2n+I C are then
R+ = {Ls + Lj};<ju {L, - L,}f<j
whereas in the case of so2nC we have
The primitive positive roots are
272
18...
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...
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...
Even though so6C is isomorphic to a Lie algebra we have already examined,
it is worth going through the analysis of its representations for what amounts
to a second time, partly so as to understand the isomorphism better, but
mainly because we will see clearly in the case of so6C a number of phenomena
that will hold true of the even orthogonal groups in general...
Note that we have now identified, in terms of tensor powers of the standard
one, irreducible representations of so6C with highest weight vectors Lu
Lt + L2 + L3 and Ll + L2 — L3 lying along the edge of the Weyl chamber,
as well as one with highest weight Lt + L2 lying in a face...
a+b=k
We also can say how each factor on the right-hand side of this expression
decomposes as a representation of sInC: we have contraction maps
Va.b- A°W ® AbW* -* A"'1 W ® A* W*;
and the kernel of HfOji is the irreducible representation Wia'b) with highest
weight 2Li + ••• + 2La + La+l +¦¦¦ + Ln_b...
(ii) Note also that by the above, AT is the direct sum of the two irreducible
representations Vlx and T2f with highest weights 2a = Ll + ¦¦¦ + Ln and
2/? = Lt +•+?„_!- Ln...
There remains the problem of constructing irreducible representations Ty
whose highest weight y involves an odd number of ce's and /?'s...
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...
?
We have thus constructed one-half of the irreducible representations of
so2n+iC: any weight y in the closed Weyl chamber can be written
y = atLt+ a2{Ll + L2) + • ¦ ¦ + ^^(L, + ¦ ¦ ¦ + ?„_,) + an(L, + • • ¦ + Ln)/2
with a, e N; and if an is even, the representation
Sym"' V ® • • • ® Sym^-'fA" V) ® Syma-/2(A"K)
will contain an irreducible representation Fy with highest weight y...
Note that at least one of each pair of associated partitions will
have a Young diagram with at most \m rows...
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...
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