pureFree -- computes a GL(V)-equivariant map whose resolution is pure, or the reduction mod p of such a map

• Usage:

  pureFree(d, P)

  pureFree(d, P, Convention => "Row" | "Filling" | "Weyl")

• Inputs:
  • d, a list, a list of degrees (increasing numbers)
  • P, a polynomial ring, a polynomial ring over a field $K$ in $n$ variables
• Optional inputs:
  • Convention => a string, default value "Row", , basis convention used internally by pieri. Defaults to "Row"; the resulting Betti table does not depend on the choice.
• Outputs:
  • a matrix, A map whose cokernel has Betti diagram with degree sequence $d$ if $K$ has characteristic 0. If $K$ has positive characteristic $p$, then the corresponding map is calculated over $\mathbb{Q}$ and is lifted to a $\mathbb{Z}$-form which is then reduced mod $p$.

i1 : betti res coker pureFree({0,1,2,4}, QQ[a,b,c]) -- degree sequence {0,1,2,4}

            0 1 2 3
o1 = total: 3 8 6 1
         0: 3 8 6 .
         1: . . . 1

o1 : BettiTally

i2 : betti res coker pureFree({0,1,2,4}, ZZ/2[a,b,c]) -- same map, but reduced mod 2

            0 1 2 3
o2 = total: 3 8 6 1
         0: 3 8 6 .
         1: . . . 1

o2 : BettiTally

i3 : betti res coker pureFree({0,1,2,4}, GF(4)[a,b,c]) -- can also use non prime fields

            0 1 2 3
o3 = total: 3 8 6 1
         0: 3 8 6 .
         1: . . . 1

o3 : BettiTally

i4 : -- The Betti table is independent of the chosen convention:
     P = QQ[a,b,c];

i5 : bettiR = betti res coker pureFree({0,1,2,4}, P, Convention => "Row");

i6 : bettiF = betti res coker pureFree({0,1,2,4}, P, Convention => "Filling");

i7 : bettiR == bettiF

o7 = true

i8 :