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Schubert2 :: symmetricPower(RingElement,AbstractSheaf)

symmetricPower(RingElement,AbstractSheaf) -- symmetric power of an abstract sheaf

Synopsis

Description

In the first example, we let n be a natural number.
i1 : tangentBundle PP^4

o1 = a sheaf

o1 : an abstract sheaf of rank 4 on a flag bundle
i2 : symmetricPower_4 oo

o2 = a sheaf

o2 : an abstract sheaf of rank 35 on a flag bundle
i3 : chern oo

o3 = 1 + 175H    + 14910H    + 823970H    + 33198935H
             2,1         2,2          2,3            2,4

                     QQ[][h, H   , H   , H   , H   ]
                              2,1   2,2   2,3   2,4
o3 : ---------------------------------------------------------------
     (h + H   , h*H    + H   , h*H    + H   , h*H    + H   , h*H   )
           2,1     2,1    2,2     2,2    2,3     2,3    2,4     2,4
In the next example, we let n be a free parameter in the “intersection ring” of the base variety.
i4 : pt = base n

o4 = pt

o4 : an abstract variety of dimension 0
i5 : X = projectiveSpace'_2 pt

o5 = X

o5 : a flag bundle with ranks {2, 1}
i6 : tangentBundle X

o6 = a sheaf

o6 : an abstract sheaf of rank 2 on X
i7 : F = symmetricPower_n oo

o7 = F

o7 : an abstract sheaf of rank n + 1 on X
i8 : chern F

          3 2   3       9 4   5 3   3 2   1   2
o8 = 1 + (-n  + -n)h + (-n  + -n  + -n  + -n)h
          2     2       8     4     8     4

           QQ[n][H   , H   , h]
                  1,1   1,2
o8 : -------------------------------
     (H    + h, H    + H   h, H   h)
       1,1       1,2    1,1    1,2
i9 : ch F

                3 2   3        3   3 2   1   2
o9 = (n + 1) + (-n  + -n)h + (n  + -n  - -n)h
                2     2            4     4

           QQ[n][H   , H   , h]
                  1,1   1,2
o9 : -------------------------------
     (H    + h, H    + H   h, H   h)
       1,1       1,2    1,1    1,2
i10 : chi F

       3     2
o10 = n  + 3n  + 3n + 1

o10 : QQ[n]

See also