This page provides representation theoretic information about double covers of even rank (noncompact) spin groups of type Bn. We consider only half integral infinitesimal character &lambda = &lambda(p,q), where
```p + q = n
p = # of integer coordinates
q = # of half-integer coordinates
```

Example
Here is a portion of the output for Spin(7,2) at &lambda(0,4):

Spin(7,2) Double Cover
Infinitesimal characater: 0 integer(s) 4 half-integer(s)
Atlas K\G/B: kgb
Number of Blocks: 1
Block Size: 16

Block 0
Infinitesimal Character (3.5, 2.5, 1.5, 0.5)

``` 0(14):    0    0    1    0     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [ic,ic,C+,rn]   1
1(19):    1    4    0    1     ( *, *)   ( *, *)   ( *, *)   ( 2, 3)     [ic,C+,C-,i2]   2
2(22):    2    5    2    3     ( *, *)   ( *, *)   ( *, *)   ( 1, *)     [ic,C+,rn,r2]   3
3(22):    3    6    3    2     ( *, *)   ( *, *)   ( *, *)   ( 1, *)     [ic,C+,rn,r2]   3
4(23):    7    1    4    4     ( *, *)   ( *, *)   ( *, *)   ( 5, 6)     [C+,C-,ic,i2]   3
5(28):   10    2    8    6     ( *, *)   ( *, *)   ( *, *)   ( 4, *)     [C+,C-,C+,r2]   4
6(28):   11    3    9    5     ( *, *)   ( *, *)   ( *, *)   ( 4, *)     [C+,C-,C+,r2]   4
7(29):    4    7    7    7     ( *, *)   ( *, *)   ( *, *)   (10,11)     [C-,ic,ic,i2]   4
8(34):   12    8    5    8     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [C+,rn,C-,ic]   5
9(34):   13    9    6    9     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [C+,rn,C-,ic]   5
10(35):    5   10   12   11     ( *, *)   ( *, *)   ( *, *)   ( 7, *)     [C-,ic,C+,r2]   5
11(35):    6   11   13   10     ( *, *)   ( *, *)   ( *, *)   ( 7, *)     [C-,ic,C+,r2]   5
12(38):    8   14   10   12     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [C-,C+,C-,ic]   6
13(38):    9   15   11   13     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [C-,C+,C-,ic]   6
14(41):   14   12   14   14     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [rn,C-,ic,ic]   7
15(41):   15   13   15   15     ( *, *)   ( *, *)   ( *, *)   ( *, *)     [rn,C-,ic,ic]   7
```
• Kgb Graph
• Block Text
• Block Graph
• R-Polynomial List
• R-Polynomial Basis
• KL-Polynomial List
• KL-Polynomial Basis

The output begins with some basic group and block information as well as a link to the K\G/B graph. This is the full K\G/B graph as generated by Atlas. The edges correspond to cross-actions and Cayley transforms in simple roots and are colored as follows:

```blue - type I Cayley transform
green - type II Cayley transform
black - complex cross action
```
Next is an ASCII printout of the structure a block at the various infinitesimal characters (only one is shown here). This is similar to the output of the Atlas 'block' command. The numbers in parentheses on the left represent the K\G/B orbit number corresponding to each block element. NOTE: The output of this section is supressed if the size of the block is too large.

Finally there are several links to more information about this particular block. Here is a brief explanation of each link:

• The Kgb Graph link is to a subgraph of the full K\G/B graph showing only those orbits that support representations at &lambda and only cross actions in integral complex roots.
• The Block Text link is to a text file containing the same ASCII block output as shown above.
• The Block Graph link is to a graph for this block colored as follows
```blue - short type I Cayley transform
red - long type I Cayley transform
green - type II Cayley transform
black - complex cross action
```
• The final four items are links to text files containing lists of the R-polynomials and Kazhdan-Lusztig polynomials for this block. The output is similar to the Atlas format.
• Remark
In general, there are two distinct choices for the genuine action of the center and the blocks represented here are for a fixed choice of central character. For a given &lambda(p,q), a group will often possess blocks of genuine representations at both central characters. However, the block at &lambda(p,q) for one central character is isomorphic to the block with the other central character at &lambda(q,p). In particular, no new information is obtained by considering both central characters.

For each group below, only the &lambda(p,q)s leading to nonzero block sizes are listed. Follow the associated links to see the corresponding block data.

### Rank 2

G=Spin(4,1)
• p=1 q=1
Example: &lambda = (1, 1/2)
Block Size: 1

• p=2 q=0
Example: &lambda = (2, 1)
Block Size: 4
G=Spin(3,2)
• p=0 q=2
Example: &lambda = (3/2, 1/2)
Block Size: 4

• p=1 q=1
Example: &lambda = (1, 1/2)
Block Size: 9

• p=2 q=0
Example: &lambda = (2, 1)
Block Size: 2

### Rank 4

G=Spin(8,1)
• p=3 q=1
Example: &lambda = (3, 2, 1, 1/2)
Block Size: 1

• p=4 q=0
Example: &lambda = (4, 3, 2, 1)
Block Size: 6
G=Spin(7,2)
• p=0 q=4
Example: &lambda = (7/2, 5/2, 3/2, 1/2)
Block Size: 16

• p=1 q=3
Example: &lambda = (3, 5/2, 3/2, 1/2)
Block Size: 21

• p=2 q=2
Example: &lambda = (3, 2, 3/2, 1/2)
Block Size: 2
G=Spin(6,3)
• p=1 q=3
Example: &lambda = (3, 5/2, 3/2, 1/2)
Block Size: 1

• p=2 q=2
Example: &lambda = (3, 2, 3/2, 1/2)
Block Size: 14

• p=3 q=1
Example: &lambda = (3, 2, 1, 1/2)
Block Size: 38

• p=4 q=0
Example: &lambda = (4, 3, 2, 1)
Block Size: 16
G=Spin(5,4)
• p=0 q=4
Example: &lambda = (7/2, 5/2, 3/2, 1/2)
Block Size: 6

• p=1 q=3
Example: &lambda = (3, 5/2, 3/2, 1/2)
Block Size: 38

• p=2 q=2
Example: &lambda = (3, 2, 3/2, 1/2)
Block Size: 72

• p=3 q=1
Example: &lambda = (3, 2, 1, 1/2)
Block Size: 21

• p=4 q=0
Example: &lambda = (4, 3, 2, 1)
Block Size: 2

### Rank 6

G=Spin(12,1)
• p=5 q=1
Example: &lambda = (5, 4, 3, 2, 1, 1/2)
Block Size: 1

• p=6 q=0
Example: &lambda = (6, 5, 4, 3, 2, 1)
Block Size: 8

G=Spin(11,2)
• p=0 q=6
Example: &lambda = (11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 36

• p=1 q=5
Example: &lambda = (5, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 33

• p=2 q=4
Example: &lambda = (5, 4, 7/2, 5/2, 3/2, 1/2)
Block Size: 2

G=Spin(10,3)
• p=3 q=3
Example: &lambda = (5, 4, 3, 5/2, 3/2, 1/2)
Block Size: 1

• p=4 q=2
Example: &lambda = (5, 4, 3, 2, 3/2, 1/2)
Block Size: 24

• p=5 q=1
Example: &lambda = (5, 4, 3, 2, 1, 1/2)
Block Size: 87

• p=6 q=0
Example: &lambda = (6, 5, 4, 3, 2, 1)
Block Size: 50

G=Spin(9,4)
• p=0 q=6
Example: &lambda = (11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 50

• p=1 q=5
Example: &lambda = (5, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 215

• p=2 q=4
Example: &lambda = (5, 4, 7/2, 5/2, 3/2, 1/2)
Block Size: 276

• p=3 q=3
Example: &lambda = (5, 4, 3, 5/2, 3/2, 1/2)
Block Size: 49

• p=4 q=2
Example: &lambda = (5, 4, 3, 2, 3/2, 1/2)
Block Size: 2

G=Spin(8,5)
• p=1 q=5
Example: &lambda = (5, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 1

• p=2 q=4
Example: &lambda = (5, 4, 7/2, 5/2, 3/2, 1/2)
Block Size: 24

• p=3 q=3
Example: &lambda = (5, 4, 3, 5/2, 3/2, 1/2)
Block Size: 207

• p=4 q=2
Example: &lambda = (5, 4, 3, 2, 3/2, 1/2)
Block Size: 466

• p=5 q=1
Example: &lambda = (5, 4, 3, 2, 1, 1/2)
Block Size: 215

• p=6 q=0
Example: &lambda = (6, 5, 4, 3, 2, 1)
Block Size: 36

G=Spin(7,6)
• p=0 q=6
Example: &lambda = (11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 8

• p=1 q=5
Example: &lambda = (5, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 87

• p=2 q=4
Example: &lambda = (5, 4, 7/2, 5/2, 3/2, 1/2)
Block Size: 466

• p=3 q=3
Example: &lambda = (5, 4, 3, 5/2, 3/2, 1/2)
Block Size: 799

• p=4 q=2
Example: &lambda = (5, 4, 3, 2, 3/2, 1/2)
Block Size: 276

• p=5 q=1
Example: &lambda = (5, 4, 3, 2, 1, 1/2)
Block Size: 33

• p=6 q=0
Example: &lambda = (6, 5, 4, 3, 2, 1)
Block Size: 2

### Rank 8

G=Spin(16,1)
• p=7 q=1
Example: &lambda = (7, 6, 5, 4, 3, 2, 1, 1/2)
Block Size: 1

• p=8 q=0
Example: &lambda = (8, 7, 6, 5, 4, 3, 2, 1)
Block Size: 10

G=Spin(15,2)
• p=0 q=8
Example: &lambda = (15/2, 13/2, 11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 64

• p=1 q=7
Example: &lambda = (7, 13/2, 11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 45

• p=2 q=6
Example: &lambda = (7, 6, 11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 2

G=Spin(14,3)
• p=5 q=3
Example: &lambda = (7, 6, 5, 4, 3, 5/2, 3/2, 1/2)
Block Size: 1

• p=6 q=2
Example: &lambda = (7, 6, 5, 4, 3, 2, 3/2, 1/2)
Block Size: 34

• p=7 q=1
Example: &lambda = (7, 6, 5, 4, 3, 2, 1, 1/2)
Block Size: 156

• p=8 q=0
Example: &lambda = (8, 7, 6, 5, 4, 3, 2, 1)
Block Size: 112

G=Spin(13,4)
• p=0 q=8
Example: &lambda = (15/2, 13/2, 11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 196

• p=1 q=7
Example: &lambda = (7, 13/2, 11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 644

• p=2 q=6
Example: &lambda = (7, 6, 11/2, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 616

• p=3 q=5
Example: &lambda = (7, 6, 5, 9/2, 7/2, 5/2, 3/2, 1/2)
Block Size: 77

• p=4 q=4
Example: &lambda = (7, 6, 5, 4, 7/2, 5/2, 3/2, 1/2)
Block Size: 2