bases -- gives a list of the bases of the different graded pieces of the ExteriorExtension

Description

Let's build a 7-graded algebra and see the bases of each piece

i1 : extensionAlg = exteriorExtension(3, 7);

i2 : extensionAlg.bases#0

o2 = {| 1 0  0 0 0 0 0 |, | 0 0 0  0 0 0 0 |, | 0 0 0 0  0 0 0 |, | 0 0 0 0
      | 0 -1 0 0 0 0 0 |  | 0 1 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0
      | 0 0  0 0 0 0 0 |  | 0 0 -1 0 0 0 0 |  | 0 0 1 0  0 0 0 |  | 0 0 0 0
      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 -1 0 0 0 |  | 0 0 0 1
      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0
      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0
      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0
     ------------------------------------------------------------------------
     0  0 0 |, | 0 0 0 0 0 0  0 |, | 0 0 0 0 0 0 0  |, | 0 1 0 0 0 0 0 |, | 0
     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0
     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0
     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0
     -1 0 0 |  | 0 0 0 0 1 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0
     0  0 0 |  | 0 0 0 0 0 -1 0 |  | 0 0 0 0 0 1 0  |  | 0 0 0 0 0 0 0 |  | 0
     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 -1 |  | 0 0 0 0 0 0 0 |  | 0
     ------------------------------------------------------------------------
     0 1 0 0 0 0 |, | 0 0 0 1 0 0 0 |, | 0 0 0 0 1 0 0 |, | 0 0 0 0 0 1 0 |,
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     ------------------------------------------------------------------------
     | 0 0 0 0 0 0 1 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 1 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     ------------------------------------------------------------------------
     |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0
     |  | 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 1
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     ------------------------------------------------------------------------
     0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     ------------------------------------------------------------------------
     0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     ------------------------------------------------------------------------
     0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |,
     0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 | 
     ------------------------------------------------------------------------
     | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0
     ------------------------------------------------------------------------
     |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0 |  | 0 0 0 0 0
     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0
     ------------------------------------------------------------------------
     0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0
     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 1
     ------------------------------------------------------------------------
     0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0
     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0
     ------------------------------------------------------------------------
     0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |}
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
     0 0 0 1 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 |  | 0 0 0 0 0 1 0 |

o2 : List

i3 : extensionAlg.bases#1

o3 = {e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e ,
       0 1 2   0 1 3   0 1 4   0 1 5   0 1 6   0 2 3   0 2 4   0 2 5   0 2 6 
     ------------------------------------------------------------------------
     e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e ,
      0 3 4   0 3 5   0 3 6   0 4 5   0 4 6   0 5 6   1 2 3   1 2 4   1 2 5 
     ------------------------------------------------------------------------
     e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e ,
      1 2 6   1 3 4   1 3 5   1 3 6   1 4 5   1 4 6   1 5 6   2 3 4   2 3 5 
     ------------------------------------------------------------------------
     e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e }
      2 3 6   2 4 5   2 4 6   2 5 6   3 4 5   3 4 6   3 5 6   4 5 6

o3 : List

i4 : extensionAlg.bases#3

o4 = {e e , e e , e e , e e , e e , e e , e e , e e , e e , e e , e e , e e ,
       0 1   0 2   0 3   0 4   0 5   0 6   1 2   1 3   1 4   1 5   1 6   2 3 
     ------------------------------------------------------------------------
     e e , e e , e e , e e , e e , e e , e e , e e , e e }
      2 4   2 5   2 6   3 4   3 5   3 6   4 5   4 6   5 6

o4 : List

For the programmer

The object bases is a symbol.

The source of this document is in ExteriorExtensions.m2:797:0.