--- Computations for comparing degenerations of Gr(3,6) from plabic graphs with --- the tropical Grassmannian Trop Gr(3,6). --- Written by Milena Hering, with help from Arend Bayer, Daniel Erman, Claudiu Raicu, Greg Smith. needsPackage "Polyhedra" loadPackage "gfanInterface" generateInitialIdeals = L -> ( L1 := apply(L, j -> apply (j, i -> -i + max j)); II = flatten for i from 0 to #L1-1 list ( M := {}; append(M,initialIdeal(L1#i,I')) ); II ) sgn = lis -> ( rez := 1; for i from 0 to #lis-2 do for j from i+1 to #lis-1 do if lis#i > lis#j then rez = rez * (-1); rez ) sigmaMap = (L,k,d,I) -> ( f := map(R,R,apply(apply(subsets(k+1,d+1),subs -> apply(subs,i->L#i)), ter -> p_(toSequence sort ter) * sgn(ter))); f(I) ) permutedIdeals = (k,d,I) -> ( L := permutations(k+1); for i from 0 to #L-1 list sigmaMap(L#i,k,d,I) ) -- list of weight vectors obtained from plabic graph in the order of the table in our paper PlabicGraphVectors = { {0,0,1,1,1,1,1,1,1,4,1,1,1,1,1,4,4,4,5,5}, {0,0,0,3,0,0,3,3,4,4,3,3,4,4,4,4,4,4,4,7}, {0,0,1,1,0,1,1,2,2,5,3,3,3,3,3,5,3,3,5,6}, {0,0,0,0,0,0,0,1,1,2,0,0,0,1,1,2,2,2,3,5}, {0,0,2,3,1,2,3,2,3,6,4,4,4,4,4,6,5,5,6,6}, {0,0,1,2,0,1,2,2,3,5,4,4,4,4,4,5,4,4,5,6}, {0,0,0,0,0,0,0,2,2,3,1,1,1,2,2,3,2,2,3,6}, {0,0,0,1,0,0,1,2,2,2,0,0,1,2,2,2,3,3,3,6}, {0,0,0,1,0,0,1,3,3,3,1,1,2,3,3,3,3,3,3,7}, {0,0,0,3,1,1,3,2,3,3,1,1,3,2,3,3,5,5,5,6}, {0,0,1,3,2,2,3,2,3,4,2,2,3,2,3,4,6,6,6,6}, {0,0,1,4,2,2,4,2,4,5,3,3,4,3,4,5,6,6,6,6}, {0,0,2,2,1,2,2,2,2,6,3,3,3,3,3,6,4,4,6,6}, {0,0,0,4,1,1,4,2,4,4,2,2,4,3,4,4,5,5,5,6}, {0,0,0,0,0,0,0,2,2,4,2,2,2,3,3,4,3,3,4,6}, {0,0,1,3,1,1,3,1,3,5,3,3,4,3,4,5,5,5,5,5}, {0,0,0,2,0,0,2,2,3,3,0,0,2,2,3,3,4,4,4,6}, {0,0,0,2,1,1,2,2,2,2,1,1,2,2,2,2,4,4,4,6}, {0,0,2,4,2,3,4,3,4,6,4,4,4,4,4,6,6,6,7,7}, {0,0,1,1,0,1,1,2,2,4,2,2,2,2,2,4,2,2,4,6}, {0,0,1,1,1,1,1,1,1,5,2,2,2,2,2,5,4,4,5,5}, {0,0,0,0,0,0,0,1,1,3,0,0,0,1,1,3,3,3,4,5}, {0,0,0,2,0,0,2,3,3,3,2,2,3,3,3,3,3,3,3,7}, {0,0,0,4,1,1,4,3,4,4,3,3,4,4,4,4,5,5,5,7}, {0,0,1,2,1,1,2,1,2,4,1,1,2,1,2,4,5,5,5,5}, {0,0,1,3,0,1,3,3,4,5,4,4,4,4,4,5,4,4,5,7}, {0,0,1,3,1,2,3,3,3,4,3,3,3,3,3,4,4,4,5,7}, {0,0,0,1,0,0,1,1,2,4,1,1,2,2,3,4,4,4,4,5}, {0,0,0,0,0,0,0,1,1,4,1,1,1,2,2,4,3,3,4,5}, {0,0,1,2,1,1,2,1,2,5,2,2,3,2,3,5,5,5,5,5}, {0,0,0,1,0,0,1,1,2,3,0,0,1,1,2,3,4,4,4,5}, {0,0,1,2,0,1,2,3,3,4,3,3,3,3,3,4,3,3,4,7}, {0,0,0,3,1,1,3,3,3,3,2,2,3,3,3,3,4,4,4,7}, {0,0,1,4,1,2,4,3,4,5,4,4,4,4,4,5,5,5,6,7} } -- list of weight vectors given in Speyer-Sturmfels in the interior of the 3d cones in Gr(3,6) -- in order FFGG EEEE EEFF1 EEFF2 EFFG EEEG EEFG TropGrassmannian3d={{1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2}, {1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2}, {0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 2, 1, 1, 1, 1}, {0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 1, 2, 1}, {0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 1, 1}} -- list of weight vectors given in Speyer-Sturmfels in the interior of some 2d cones in trop Gr(3,6) -- in order EEE,EEF, EEG, EFF, EFG, FFG, FGG TropGrassmannian2d={{0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0}, {0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,2,0,1,0,0}, {1,1,1,1,0,0,1,0,0,0,0,0,2,0,0,0,1,0,0,0}, {0,0,1,1,0,2,0,1,0,1,0,0,1,0,1,1,0,0,0,0}, {0,0,1,0,0,1,3,1,0,1,0,0,1,0,0,0,0,1,2,0}, {1,0,2,0,0,2,0,2,1,2,1,1,1,0,0,0,0,0,0,1}, {1,1,1,1,0,0,1,2,0,0,0,0,1,2,0,0,2,1,1,2}} -- list of weight vectors given in Speyer-Sturmfels in the interior of some 1d cones in trop Gr(3,6) -- in order EE, FF, GG, EF1, EF2, EG, FG TropGrassmannian1d = {{1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0}, {1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0}, {1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,1,1,1,1,1}, {2,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1}, {1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0}, {2,2,2,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0}} -- list of weight vectors given in Speyer-Sturmfels that are vertices in trop Gr(3,6) -- E, F, G TropGrassmannian0d = {{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0}} -- list of weight vectors given in Speyer-Sturmfels TropGrassmannian = TropGrassmannian3d|TropGrassmannian2d|TropGrassmannian1d|TropGrassmannian0d I=Grassmannian(2,5) R= QQ**ring(I) I' = substitute(I,R) printWidth = 70 PlabicIdeals = generateInitialIdeals PlabicGraphVectors TropicalIdeals = generateInitialIdeals TropGrassmannian equalsTropicalIdeal = I -> ( for i from 0 to #TropicalIdeals-1 list ( if (I==TropicalIdeals#i) == true then return i else continue); return -1 ) end ---------------------------- restart load "permutations.m2" k=5 d=2 use R Permutations=permutations(k+1); ----This computes a list. Each entry is a list of 3 elements. ----The first one is the index of the plabic graph vector from PlabicGraphVectors, ----the second one is a permutaion, ----the third one is the index of a vector from the tropical Grassmannian as in TropGrassmannian. ----If such a triple is in the list, it means that the initial ideal with respect to the Plabic graph vector ----is equal to the initial ideal with respect to the Tropical Grassmannaian vector after applying the ----permutation. M = {} for l from 0 to #PlabicIdeals-1 do ( P=permutedIdeals(k,d,PlabicIdeals#l); for i from 0 to #P-1 do ( j = equalsTropicalIdeal(P#i); if (j != -1) then M = append(M, {l,Permutations#i, j}))) M