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PieriMaps :: pieri

pieri -- computes a matrix representation for a Pieri inclusion of representations of a general linear group

Synopsis

Description

Convention: the partition (d) represents the dth symmetric power, while the partition (1,...,1) represents the dth exterior power. Using the notation from the output, mu/lambda must be a horizontal strip. Precisely, this means that lambda_i >= mu_(i+1) for all i. If this condition is not satisfied, the program throws an error because a nonzero equivariant map of the desired form will not exist.
i1 : pieri({3,1}, {1}, QQ[a,b,c]) -- removes the last box from row 1 of the partition {3,1}

o1 = | 3a 0  b  0  c  0  0  0  0     0 0   0  0  0  0  |
     | 0  3a 0  b  0  c  0  0  0     0 0   0  0  0  0  |
     | 0  0  2a 0  0  0  2b 0  c     0 0   0  0  0  0  |
     | 0  0  0  2a 0  0  0  2b 0     c 0   0  0  0  0  |
     | 0  0  0  0  2a 0  0  0  b     0 2c  0  0  0  0  |
     | 0  0  0  0  0  2a 0  0  0     b 0   2c 0  0  0  |
     | 0  0  0  0  0  0  0  a  -1/2a 0 0   0  3b c  0  |
     | 0  0  0  0  0  0  0  0  0     a -2a 0  0  2b 2c |

                         8                   15
o1 : Matrix (QQ[a, b, c])  <--- (QQ[a, b, c])
i2 : res coker oo -- resolve this map

                  8                  15                  10                  3
o2 = (QQ[a, b, c])  <-- (QQ[a, b, c])   <-- (QQ[a, b, c])   <-- (QQ[a, b, c])  <-- 0
                                                                                    
     0                  1                   2                   3                  4

o2 : ChainComplex
i3 : betti oo -- check that the resolution is pure

            0  1  2 3
o3 = total: 8 15 10 3
         0: 8 15  . .
         1: .  . 10 .
         2: .  .  . 3

o3 : BettiTally

See also

Ways to use pieri :