Blocks of RepresentationsΒΆ

Recall that blocks of a representation are where the Kazshdan-Lusztig polynomials live. So it is vey useful to find the block of a single representation.

Let \(G=Sp(4, \mathbb R)\) and let us find the block of the trivial representation

atlas> G:Sp(4,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas>
atlas> set t=trivial(G)
Variable t: Param (overriding previous instance, which had type Param)
atlas> print_block(t)
Parameter defines element 10 of the following block:
  0:  0  [i1,i1]   1   2   ( 4, *)  ( 5, *)  *(x= 0,lam_rho=  [0,0], nu=  [0,0]/1)  e
  1:  0  [i1,i1]   0   3   ( 4, *)  ( 6, *)  *(x= 1,lam_rho=  [0,0], nu=  [0,0]/1)  e
  2:  0  [ic,i1]   2   0   ( *, *)  ( 5, *)  *(x= 2,lam_rho=  [0,0], nu=  [0,0]/1)  e
  3:  0  [ic,i1]   3   1   ( *, *)  ( 6, *)  *(x= 3,lam_rho=  [0,0], nu=  [0,0]/1)  e
  4:  1  [r1,C+]   4   9   ( 0, 1)  ( *, *)  *(x= 4,lam_rho=  [0,0], nu= [1,-1]/2)  1^e
  5:  1  [C+,r1]   7   5   ( *, *)  ( 0, 2)  *(x= 5,lam_rho=  [0,0], nu=  [0,1]/1)  2^e
  6:  1  [C+,r1]   8   6   ( *, *)  ( 1, 3)  *(x= 6,lam_rho=  [0,0], nu=  [0,1]/1)  2^e
  7:  2  [C-,i1]   5   8   ( *, *)  (10, *)  *(x= 7,lam_rho=  [0,0], nu=  [2,0]/1)  1x2^e
  8:  2  [C-,i1]   6   7   ( *, *)  (10, *)  *(x= 8,lam_rho=  [0,0], nu=  [2,0]/1)  1x2^e
  9:  2  [i2,C-]   9   4   (10,11)  ( *, *)  *(x= 9,lam_rho=  [0,0], nu=  [3,3]/2)  2x1^e
 10:  3  [r2,r1]  11  10   ( 9, *)  ( 7, 8)  *(x=10,lam_rho=  [0,0], nu=  [2,1]/1)  1^2x1^e
 11:  3  [r2,rn]  10  11   ( 9, *)  ( *, *)  *(x=10,lam_rho=  [1,1], nu=  [2,1]/1)  1^2x1^e
 atlas>

This is the block of the trivial representation, which as it says above, it is parameter number 10 in this block, whith real roots labeled r2 and r1 in the tau invariant. This block has two principal series in it. Namely the two with x=10. This block has 12 representations and we have seen that there are 18 representations with infinitesimal character rho. it turns out they are divided into blocks. The other principal series are in other blocks. Let us find them

set params=all_parameters_gamma(G,rho(G))
Variable params: [Param]
atlas> print_block(params[17])
Parameter defines element 10 of the following block:
 0:  0  [i1,i1]   1   2   ( 4, *)  ( 5, *)  *(x= 0,lam_rho=  [0,0], nu=  [0,0]/1)  e
 1:  0  [i1,i1]   0   3   ( 4, *)  ( 6, *)  *(x= 1,lam_rho=  [0,0], nu=  [0,0]/1)  e
 2:  0  [ic,i1]   2   0   ( *, *)  ( 5, *)  *(x= 2,lam_rho=  [0,0], nu=  [0,0]/1)  e
 3:  0  [ic,i1]   3   1   ( *, *)  ( 6, *)  *(x= 3,lam_rho=  [0,0], nu=  [0,0]/1)  e
 4:  1  [r1,C+]   4   9   ( 0, 1)  ( *, *)  *(x= 4,lam_rho=  [0,0], nu= [1,-1]/2)  1^e
 5:  1  [C+,r1]   7   5   ( *, *)  ( 0, 2)  *(x= 5,lam_rho=  [0,0], nu=  [0,1]/1)  2^e
 6:  1  [C+,r1]   8   6   ( *, *)  ( 1, 3)  *(x= 6,lam_rho=  [0,0], nu=  [0,1]/1)  2^e
 7:  2  [C-,i1]   5   8   ( *, *)  (11, *)  *(x= 7,lam_rho=  [0,0], nu=  [2,0]/1)  1x2^e
 8:  2  [C-,i1]   6   7   ( *, *)  (11, *)  *(x= 8,lam_rho=  [0,0], nu=  [2,0]/1)  1x2^e
 9:  2  [i2,C-]   9   4   (10,11)  ( *, *)  *(x= 9,lam_rho=  [0,0], nu=  [3,3]/2)  2x1^e
10:  3  [r2,rn]  11  10   ( 9, *)  ( *, *)  *(x=10,lam_rho=  [1,1], nu=  [2,1]/1)  1^2x1^e
11:  3  [r2,r1]  10  11   ( 9, *)  ( 7, 8)  *(x=10,lam_rho=  [0,0], nu=  [2,1]/1)  1^2x1^e
atlas>

This is again the block of the trivial

atlas> print_block(params[16])
Parameter defines element 0 of the following block:
0:  0  [rn,rn]  0  0   (*,*)  (*,*)  *(x=10,lam_rho=  [0,1], nu=  [2,1]/1)  1^2x1^e
atlas>
atlas> params[16]
Value: final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1)
atlas>

This is another principal series, all by itself in another block because it is an irreducible principal series. Let us find the other one

atlas> print_block(params[15])
Parameter defines element 4 of the following block:
0:  0  [C+,rn]  2  0   (*,*)  (*,*)  *(x= 5,lam_rho=  [0,1], nu=  [0,1]/1)  2^e
1:  0  [C+,rn]  3  1   (*,*)  (*,*)  *(x= 6,lam_rho=  [0,1], nu=  [0,1]/1)  2^e
2:  1  [C-,i1]  0  3   (*,*)  (4,*)  *(x= 7,lam_rho=  [1,0], nu=  [2,0]/1)  1x2^e
3:  1  [C-,i1]  1  2   (*,*)  (4,*)  *(x= 8,lam_rho=  [1,0], nu=  [2,0]/1)  1x2^e
4:  2  [rn,r1]  4  4   (*,*)  (2,3)  *(x=10,lam_rho=  [1,0], nu=  [2,1]/1)  1^2x1^e
atlas> params[15]
Value: final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1)
atlas>

This is a smaller block but it has the last 5 of the representations with infinitesimal character rho.

So, there are three blocks of sizes 1, 5 and 12 each. In fact we can get block sizes information doing the following

atlas> block_sizes (Sp(4,R))
Value:
| 0, 0,  1 |
| 0, 0,  4 |
| 1, 5, 12 |

This matrix gives information about the block sizes of all the real forms of \(Sp(4,\mathbb R)\). The last row gives the block sizes for the split group.

Recall that a block is a singleton if and only if the representation is irreducible.

Now do something similar for the real forms of E8

atlas> void: for H in real_forms(G) do prints(H) od
compact connected real group with Lie algebra 'e8'
connected real group with Lie algebra 'e8(e7.su(2))'
connected split real group with Lie algebra 'e8(R)'
atlas>

We have three real forms, the compact one, an intermediate one and the split one

atlas> block_sizes(split_form(E8))
Value:
| 0,     0,      1 |
| 0,  3150,  73410 |
| 1, 73410, 453060 |

atlas>

Again the last row gives the block sizes of the blocks of the split real form of E8. There is only one irreducible principal series and 255 reducible ones. 120 are in the second block and 135 in the other block.

As an exercise it is interesting to print each block.