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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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their representation theory will be completely elementary...
With these choices, we may take as Cartan subalgebra—in both the even
and odd cases—the subalgebra of matrices diagonal in this representation.1
' Note that if we had taken the simpler choice of Q, with M the identity matrix, the Lie algebra
would have consisted of skew-symmetric matrices, and there would have been no nonzero
diaeonal matrices in the Lie aleebra...
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...
Clearly the
action on P(U ® W) will preserve the points corresponding to decomposable
tensors (that is, points of the form [u ® w]); but the locus of such points is
just a quadric hypersurface, giving us the inverse inclusion of PGL2C x
PGL2C in PSO4C...
G = GB,V)
Note one more aspect of this example: as in the case of so3C S sl2C, the
weights of the standard representation of so4C do not generate the weight
lattice, but rather a sublattice Z{Lr, L2} of index 2 in Aw...
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...
In a similar fashion, the variety
of 2-planes lying on a quadric hypersurface in P5 turns out to be disconnected,
consisting of two components that, under the Pliicker embedding of GC,6)
in P(A3C6) = P19, span two complementary 9-planes PWX and PW2; these
two planes give the direct sum decomposition of A3 Fas an so6C-module...
The proof of part (ii) requires only one further step: we have to check the
vectors w(o>l>) with a + ft = k = n to see if any of them might be highest weight
vectors for so2nC...
We see from the above
that once we exhibit the two representations Fa and F^, we will have con-
constructed all the representations of so2nC...
Finally, we should say that the subject of the spin
representations of somC is a very rich one, and one that accommodates many different
points of view; the reader who is interested is encouraged to try some of the other
approaches that may be found in the literature...
In fact, the Clifford algebra1 will be defined below to be the associative algebra
generated by V and subject to the equation v v = Q{v, v)...
This gives an explicit formula for the bracket on /\2V:
la a b, c a </] = 2Q(b, c)a a d - 2Q{b, d)a a c
- 2Q(a, d)c a b + 2Q(a, c)d a b...
Frequently, especially when we speak of the even and odd
cases together, we call them all simply "spin representations." Elements of S
are called spinors...
To complete the proof, we must check that Spin(Q) is connected or, equiva-
lently, that the two elements in the kernel of p can be connected by a path...
One uses the real Clifford algebra Cliff(Rm, Q) associated to
the real quadratic form Q = — Qm, where Qm is the standard positive definite
quadratic form on Um...
Given A e Q+, A itself is a 3-plane in Q, and {F e Q~: F n A is a 2-plane}
is a 3-plane in Q~...
Exercise 20.49) translates to the "local triality"
equation
<&(Xv, s, t) + ®{v, Ys, t) + ®{v, s, Zt) = 0
« 2.1,22
PART IV
LIE THEORY
The purpose of this final part of the book is threefold...
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