The "small" commands are: smallkgb, smallblock, smalldualkgb and
smalldualblock. Each of these performs a piece of the corresponding
basic command kgb, block, dualkgb and dualblock.
Choose a real form of G, with corresponding (complexified) maximal
compact subgroups K. The kgb command lists the orbits of K on
G/B, numbered 0,..,m.
Choose a real form of G^v, with corresponding (complexified) maximal
compact subgroups K^v. The dualkgb command lists the orbits of K^v on
G^v/B^v, numbered 0,..,m.
The choice of real forms of G and G^v determines a block B of
representations of the given real form of G. The block is parametrized
by a subset of elements (x,y) with x<=m, y<=n. See the help file for
the block command. In general not all x or y occur.
The small commands each take as arguments real forms of G and
G^v. They compute only the elements needed for the corresponding
block. Thus:
smallkgb computes and displays only the kgb elements x arising as the
first coordinate of an element (x,y) of B
smalldualkgb computes and displays only the kgb elements y arising as
the second coordinate of an element (x,y) of B
smallblock displays the block. The list of representations is the same
as in the output of block, but (see below) the number of the elements
(x,y) may be different. Similarly for smalldualblock. These
computations are less intensive than block and dualblock, since they
only compute the kgb/dualkgb elements needed for the block.
Warning: the elements of smallkgb are numbered consecutively starting
at 0. These numbers do NOT coincide with the number of elements of the
kgb command. This holds as well for small block, and the dual
versions.
Example: Here is the block of Sp(4,R) dual to the compact group
SO(5). This block has one element.
First we run kgb and dualkgb. Note that element #10 in the output of
kgb is the same as element #0 in the output of smallkgb.
Then we run block and small block. The only difference in the output
is x=10 versus x=0.
block: type
Lie type: C2 sc s
main: kgb
(weak) real forms are:
0: sp(2)
1: sp(1,1)
2: sp(4,R)
enter your choice: 2
kgbsize: 11
Name an output file (return for stdout, ? to abandon):
0: 0 0 [n,n] 1 2 4 5 e
1: 0 0 [n,n] 0 3 4 6 e
2: 0 0 [c,n] 2 0 * 5 e
3: 0 0 [c,n] 3 1 * 6 e
4: 1 1 [r,C] 4 9 * * 1
5: 1 2 [C,r] 7 5 * * 2
6: 1 2 [C,r] 8 6 * * 2
7: 2 2 [C,n] 5 8 * 10 1,2,1
8: 2 2 [C,n] 6 7 * 10 1,2,1
9: 2 1 [n,C] 9 4 10 * 2,1,2
10: 3 3 [r,r] 10 10 * * 2,1,2,1
real: smallkgb
possible (weak) dual real forms are:
0: so(5)
1: so(4,1)
2: so(3,2)
enter your choice: 0
relevant Cartan classes: {3}
partial kgb size: 1
Name an output file (return for stdout, ? to abandon):
0: 3 3 [r,r] 0 0 * * 2,1,2,1
block: block
Name an output file (return for stdout, ? to abandon):
entering block construction... K\G/B and dual generated... done
0(10,0): 3 3 [rn,rn] 0 0 (*,*) (*,*) 2,1,2,1
block: smallblock
Name an output file (return for stdout, ? to abandon):
entering block construction... K\G/B and dual generated... done
0(0,0): 3 3 [rn,rn] 0 0 (*,*) (*,*) 2,1,2,1