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Author: Sam Hopkins
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def diagram_poset(par): els = [] for i in range(len(par)): for j in range(par[i]): els += [(i,j)] return Poset([els, lambda x,y: x[0] <= y[0] and x[1] <= y[1]]) def shifted_diagram_poset(par): els = [] for i in range(len(par)): for j in range(par[i]): els += [(i,i+j)] return Poset([els, lambda x,y: x[0] <= y[0] and x[1] <= y[1]]) def rectangle(a,b): return diagram_poset([a for i in range(b)]) def shifted_staircase(n): return shifted_diagram_poset([n-i for i in range(n)]) def staircase(n): return diagram_poset([n-i for i in range(n)]) def shifted_double_staircase(n): return shifted_diagram_poset([2*n-1-2*i for i in range(n)]) def shifted_double_staircase_mod(n): return shifted_diagram_poset([2*n-2*i for i in range(n)]) def weighted_tog_vecs(P,q): L = list(P.order_ideals_lattice()) vecs = [] for p in P: vec = [0 for i in range(len(L))] for i in range(len(L)): I = L[i] if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issuperset(I): vec[i] = 1 if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issubset(I): vec[i] = -q vecs += [vec] return vecs def constant_vec(P): return [1 for I in P.order_ideals_lattice()] def antichain_vecs(P): L = list(P.order_ideals_lattice()) vecs = [] for p in P: vec = [0 for i in range(len(L))] for i in range(len(L)): I = L[i] if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issubset(I): vec[i] = 1 vecs += [vec] return vecs def antichain_space(P,q): vecs = weighted_tog_vecs(P,q) vecs += [constant_vec(P)] M = Matrix(vecs, ring=Frac(parent(q))).transpose() vecs = antichain_vecs(P) N = Matrix(vecs, ring=Frac(parent(q))).transpose() return M.column_space().intersection(N.column_space()) def antichain_space_long(P,q): vecs = weighted_tog_vecs(P,q) vecs += [constant_vec(P)] M = Matrix(vecs, ring=Frac(parent(q))).transpose() vecs = antichain_vecs(P) N = Matrix(vecs, ring=Frac(parent(q))).transpose() print(list(P)) for v in M.column_space().intersection(N.column_space()).basis(): print(N.solve_right(v)) def order_ideal_vecs(P): L = list(P.order_ideals_lattice()) vecs = [] for p in P: vec = [0 for i in range(len(L))] for i in range(len(L)): I = L[i] if p in I: vec[i] = 1 vecs += [vec] return vecs def order_ideal_space(P,q): vecs = weighted_tog_vecs(P,q) vecs += [constant_vec(P)] M = Matrix(vecs, ring=Frac(parent(q))).transpose() vecs = order_ideal_vecs(P) N = Matrix(vecs, ring=Frac(parent(q))).transpose() return M.column_space().intersection(N.column_space()) def order_ideal_space_long(P,q): vecs = weighted_tog_vecs(P,q) vecs += [constant_vec(P)] M = Matrix(vecs, ring=Frac(parent(q))).transpose() vecs = order_ideal_vecs(P) N = Matrix(vecs, ring=Frac(parent(q))).transpose() print(list(P)) for v in M.column_space().intersection(N.column_space()).basis(): print(N.solve_right(v)) #use R.<q> = PolynomialRing(QQ) to work with indeterminate q
n=5 count=0 for P in Posets(n): count=count+1 #print(count) if antichain_space(P,1).dimension() < order_ideal_space(P,1).dimension(): print("AHHH1") P.show() print(antichain_space(P,1).dimension()) print(order_ideal_space(P,1).dimension()) #if antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)).dimension() > order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)).dimension(): #print("AHHH2") #P.show()
AHHH1
3 4
n=7 count=0 for P in Posets(n): count=count+1 #print(count) if antichain_space(P,1).dimension() > order_ideal_space(P,1).dimension(): print("AHHH1") P.show() print(antichain_space(P,1).dimension()) print(order_ideal_space(P,1).dimension()) #if antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)).dimension() > order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)).dimension(): #print("AHHH2") #P.show()
AHHH1
4 3
n=4 p=0.35 trials = 100 my_flag_1 = true my_flag_2 = true for i in range(trials): print(i) P = posets.RandomPoset(n,p) #P.show() #antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)).dimension() #order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)).dimension() #print() if antichain_space(P,1).dimension() > order_ideal_space(P,1).dimension(): print("AHHH1") P.show() my_flag_1=false if antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)).dimension() > order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)).dimension(): print("AHHH2") P.show() my_flag_2=false
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 AHHH2
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
n=15 p=0.35 trials = 100 my_flag_1 = true my_flag_2 = true for i in range(trials): print(i) P = posets.RandomPoset(14,0.35) if antichain_space(P,1).dimension() > order_ideal_space(P,1).dimension(): print("AHHH1") P.show() my_flag_1=false if antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)).dimension() > order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)).dimension(): print("AHHH2") P.show() my_flag_2=false
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
P = posets.RandomPoset(14,0.35) P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 90 and dimension 3 over Rational Field Basis matrix: 3 x 90 dense matrix over Rational Field Vector space of degree 90 and dimension 4 over Rational Field Basis matrix: 4 x 90 dense matrix over Rational Field Vector space of degree 90 and dimension 1 over Rational Field Basis matrix: 1 x 90 dense matrix over Rational Field Vector space of degree 90 and dimension 1 over Rational Field Basis matrix: 1 x 90 dense matrix over Rational Field
P = rectangle(3,3) P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)) #print() #list(P.order_ideals_lattice())
Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1 -1 0 -1 -1 -1 -1 -1 -1 -2 -1] [ 0 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0] [ 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0] [ 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1] [ 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1] Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 -1 -1 0 -1 0 0 -1 0 0 -1 -1 -1 -1 -2 -1] [ 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1] [ 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1] [ 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1] [ 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1] Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1 -1 0 -1 -1 -1 -1 -1 -1 -2 -1] [ 0 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0] [ 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0] [ 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1] [ 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1] Vector space of degree 20 and dimension 2 over Rational Field Basis matrix: [0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1] [0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1]
P = shifted_staircase(3) P.show() antichain_space(P,1) order_ideal_space(P,1) print() antichain_space_long(P,1) print() print() order_ideal_space_long(P,1) print() antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)) print() list(P.order_ideals_lattice())
Vector space of degree 8 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1] [ 0 0 1 0 0 0 0 -1] [ 0 0 0 1 0 1 0 2] [ 0 0 0 0 1 1 0 2] [ 0 0 0 0 0 0 1 -1] Vector space of degree 8 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1] [ 0 0 1 0 0 -1 0 0] [ 0 0 0 1 0 1 0 2] [ 0 0 0 0 1 1 0 2] [ 0 0 0 0 0 0 1 -1] [(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2)] (1, 0, 0, 0, 0, -1) (0, 1, 0, 0, 0, -1) (0, 0, 1, 0, 0, 2) (0, 0, 0, 1, 0, 2) (0, 0, 0, 0, 1, -1) [(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2)] (1, -1, 0, 0, 0, -1) (0, 1, -1, -1, 1, 0) (0, 0, 1, 0, -1, 2) (0, 0, 0, 1, -1, 2) (0, 0, 0, 0, 1, -2) Vector space of degree 8 and dimension 4 over Rational Field Basis matrix: [ 0 1 0 0 0 0 -1 0] [ 0 0 1 0 0 0 0 -1] [ 0 0 0 1 0 1 1 1] [ 0 0 0 0 1 1 1 1] Vector space of degree 8 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 0 0 0 -1 0] [ 0 0 0 1 0 1 1 1] [ 0 0 0 0 1 1 1 1] [{}, {(0, 0)}, {(0, 1), (0, 0)}, {(0, 1), (0, 2), (0, 0)}, {(0, 1), (1, 1), (0, 0)}, {(0, 1), (0, 2), (1, 1), (0, 0)}, {(0, 1), (0, 2), (1, 2), (0, 0), (1, 1)}, {(0, 1), (0, 2), (1, 2), (2, 2), (0, 0), (1, 1)}]
P = staircase(4).dual() P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13)) #print() #list(P.order_ideals_lattice())
Vector space of degree 42 and dimension 4 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 1/2 1/2 1/2 1 1/2 0 0 1 0 0 1/2 1 1 1 0 1/2 1/2 1 1/2 1/2 1/2 0 1 1/2 1/2 1/2 1/2] [ 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1/2 1/2 1/2 1 1/2 1/2 0 1/2 1 1/2 1 0 1/2 1 1/2 1/2 1 1/2 1 0 0 1/2 1/2 1 0 1/2 0] [ 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1 1/2 1/2 1 1 1/2 1/2 0 0 0 1 1/2 0] [ 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1/2 0 1/2 1 0 1/2 1 1/2 1/2 1/2 1 1/2 1/2 0 1 1/2 1/2 1/2 1/2 1/2 1/2] Vector space of degree 42 and dimension 4 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 1/2 1/2 1/2 1 1/2 0 0 1 0 0 1/2 1 1 1 0 1/2 1/2 1 1/2 1/2 1/2 0 1 1/2 1/2 1/2 1/2] [ 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1/2 1/2 1/2 1 1/2 1/2 0 1/2 1 1/2 0 0 1/2 1 1/2 1/2 0 1/2 0 1 1 1/2 1/2 0 1 1 1/2] [ 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 0 1/2 1/2 0 0 1/2 1/2 1 1 1 0 1 1/2] [ 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1/2 0 1/2 1 0 1/2 1 1/2 1/2 1/2 1 1/2 1/2 0 1 1/2 1/2 1/2 1/2 1/2 1/2] Vector space of degree 42 and dimension 1 over Rational Field Basis matrix: [ 0 1 -1 1 -1 0 -2 0 2 0 1 -1 0 -1 1 0 -1 1 0 0 0 0 1 -1 -1 0 1 0 0 0 0 -1 1 0 1 0 -1 0 -1 1 0 0] Vector space of degree 42 and dimension 0 over Rational Field Basis matrix: []
P = shifted_double_staircase(3).dual() P.show() antichain_space(P,1) order_ideal_space(P,1) print() antichain_space_long(P,1) print() print() order_ideal_space_long(P,1) print() antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1] [ 0 0 1 0 0 1 1 1 0 0 1 1 1/2 1/2 1/2 -1/2 3/2 1/2 0 1] [ 0 0 0 1 1 0 1 1 0 1 0 1 1/2 1/2 1/2 1/2 1/2 1/2 0 1] [ 0 0 0 0 0 0 0 0 1 1 -1 -1 0 0 1 1 -1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1] [ 0 0 1 0 0 1 1 1 0 0 1 1 1/2 1/2 -1/2 1/2 1/2 3/2 0 1] [ 0 0 0 1 1 0 1 1 0 1 0 1 1/2 1/2 1/2 1/2 1/2 1/2 0 1] [ 0 0 0 0 0 0 0 0 1 1 -1 -1 0 0 1 1 -1 -1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] [(2, 2), (1, 3), (1, 2), (1, 1), (0, 4), (0, 3), (0, 2), (0, 1), (0, 0)] (1, 0, 0, 1, 0, 0, 0, 0, 1) (0, 1, 0, 1, 0, 1/2, -1/2, 0, 1) (0, 0, 0, 0, 1, 1/2, 1/2, 0, 1) (0, 0, 1, -1, 0, 0, 1, 0, -1) (0, 0, 0, 0, 0, 0, 0, 1, -1) [(2, 2), (1, 3), (1, 2), (1, 1), (0, 4), (0, 3), (0, 2), (0, 1), (0, 0)] (1, 0, -1, 1, 0, 0, 0, -1, 1) (0, 1, -1, 1, 0, -1/2, 1, -3/2, 1) (0, 0, 0, 0, 1, -1/2, 0, -1/2, 1) (0, 0, 1, -2, 0, 0, 0, 1, 0) (0, 0, 0, 0, 0, 0, 0, 1, -2) Vector space of degree 20 and dimension 2 over Rational Field Basis matrix: [ 0 1 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1] [ 0 0 1 -1 -1 1 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0] Vector space of degree 20 and dimension 1 over Rational Field Basis matrix: [0 1 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1]
P = shifted_double_staircase_mod(2).dual() P.show() antichain_space(P,1) order_ideal_space(P,1) print() antichain_space_long(P,1) print() print() order_ideal_space_long(P,1) print() antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 10 and dimension 4 over Rational Field Basis matrix: [ 0 1 0 1 0 1 1/2 1/2 0 1] [ 0 0 1 1 0 0 2/3 2/3 0 0] [ 0 0 0 0 1 1 -1/3 2/3 0 1] [ 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 10 and dimension 4 over Rational Field Basis matrix: [ 0 1 0 1 0 1 1/2 1/2 0 1] [ 0 0 1 1 0 0 2/3 -1/3 0 1] [ 0 0 0 0 1 1 -1/3 2/3 0 1] [ 0 0 0 0 0 0 0 0 1 -1] [(1, 2), (1, 1), (0, 3), (0, 2), (0, 1), (0, 0)] (0, 0, 1, 1/2, 0, 1) (1, 0, 0, 2/3, 0, 0) (0, 1, 0, -1/3, 0, 1) (0, 0, 0, 0, 1, -1) [(1, 2), (1, 1), (0, 3), (0, 2), (0, 1), (0, 0)] (0, 0, 1, -1/2, -1/2, 1) (1, -1, 0, -1/3, 1/3, 1) (0, 1, 0, -1/3, -2/3, 1) (0, 0, 0, 0, 1, -2) Vector space of degree 10 and dimension 0 over Rational Field Basis matrix: [] Vector space of degree 10 and dimension 0 over Rational Field Basis matrix: []
def half_propeller_poset(n): x={i:[i+1] for i in range(n)} y={n+1:[1]} y.update(x) return Poset(y) def propeller_poset(n): x={i:[i+1] for i in range(1,n-1)} y={n-1:[n,n+1],n:[n+2]} z={i:[i+1] for i in range(n+1,2*n)} z.update(y) z.update(x) return Poset(z) def h3_poset(): return Poset({1:[4],2:[4,5],3:[5],4:[6,7],5:[7],6:[8,9],7:[9],8:[10],9:[10,11],10:[12],11:[12],12:[13],13:[14],14:[15]}) def e6_minuscule_poset(): return Poset(([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], [[1,2],[2,3],[3,4],[4,5],[3,6],[4,7],[5,8],[6,7],[7,8],[7,9],[8,10],[9,10],[10,11],[9,12],[10,13],[11,14],[12,13],[13,14],[14,15],[15,16]]), cover_relations = True) def e7_minuscule_poset(): return Poset(([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27], [[1,2],[2,3],[3,4],[4,5],[5,6],[4,7],[5,8],[6,9],[7,8],[8,9],[8,10],[9,11],[10,11],[11,12],[10,13],[11,14],[12,15],[13,14],[14,15],[15,16],[16,17],[13,18],[14,19],[15,20],[16,21],[17,22],[18,19],[19,20],[20,21],[21,22],[21,23],[22,24],[23,24],[24,25],[25,26],[26,27]]), cover_relations = True)
R = RootSystem(['D',4]) P = R.root_poset(facade='true') P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 50 and dimension 1 over Rational Field Basis matrix: 1 x 50 dense matrix over Rational Field Vector space of degree 50 and dimension 4 over Rational Field Basis matrix: 4 x 50 dense matrix over Rational Field Vector space of degree 50 and dimension 0 over Rational Field Basis matrix: 0 x 50 dense matrix over Rational Field Vector space of degree 50 and dimension 0 over Rational Field Basis matrix: 0 x 50 dense matrix over Rational Field
R = RootSystem(['F',4]) P = R.root_poset(facade='true') P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 105 and dimension 7 over Rational Field Basis matrix: 7 x 105 dense matrix over Rational Field Vector space of degree 105 and dimension 8 over Rational Field Basis matrix: 8 x 105 dense matrix over Rational Field Vector space of degree 105 and dimension 1 over Rational Field Basis matrix: 1 x 105 dense matrix over Rational Field Vector space of degree 105 and dimension 1 over Rational Field Basis matrix: 1 x 105 dense matrix over Rational Field
R = RootSystem(['E',6]) P = R.root_poset(facade='true') P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 833 and dimension 4 over Rational Field Basis matrix: 4 x 833 dense matrix over Rational Field Vector space of degree 833 and dimension 7 over Rational Field Basis matrix: 7 x 833 dense matrix over Rational Field Vector space of degree 833 and dimension 0 over Rational Field Basis matrix: 0 x 833 dense matrix over Rational Field Vector space of degree 833 and dimension 0 over Rational Field Basis matrix: 0 x 833 dense matrix over Rational Field
P = half_propeller_poset(4) P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 8 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 1 0 0 0 2] [ 0 0 1 1 0 0 0 2] [ 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1] Vector space of degree 8 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 1 0 0 0 2] [ 0 0 1 1 0 0 0 2] [ 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1] Vector space of degree 8 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 1 0 1 0 1] [ 0 0 1 1 0 1 0 1] [ 0 0 0 0 1 -16 41 -26] Vector space of degree 8 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 1 0 1 0 1] [ 0 0 1 1 0 1 0 1] [ 0 0 0 0 1 -16 41 -26]
P = h3_poset() P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 32 and dimension 9 over Rational Field Basis matrix: [ 0 1 0 0 0 1 1 0 1 1 0 0 0 3/2 0 3/2 0 0 3/2 0 0 3/2 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 1 0 1 0 1 1/2 1 1/2 0 0 0 3/2 0 3/2 1/2 1/2 2 -1/2 -1/2 1 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 0 1 1 1 0 1/2 1 1/2 0 0 1 0 1 1 1/2 1/2 1/2 1/2 1/2 1/2 0 0 1 1 0 1 0 0 0 2] [ 0 0 0 0 0 0 0 0 0 0 1 0 1 -1 0 -1 1 0 -1 1 1 0 0 -1 1 0 -1 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 1 -1 0 1 -1 0 1 -1 0 -1 2 1 -2 0 0 0 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 1 -1 -1 1 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 1 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 -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 -1] Vector space of degree 32 and dimension 9 over Rational Field Basis matrix: [ 0 1 0 0 0 1 1 0 1 1 0 0 0 3/2 0 3/2 0 0 3/2 0 0 3/2 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 1 0 1 0 1 1/2 1 1/2 0 0 0 3/2 0 3/2 -1/2 -1/2 1 1/2 1/2 2 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 0 1 1 1 0 1/2 1 1/2 0 0 1 0 1 1 1/2 1/2 1/2 1/2 1/2 1/2 0 0 1 1 0 1 0 0 0 2] [ 0 0 0 0 0 0 0 0 0 0 1 0 1 -1 0 -1 1 0 -1 1 0 -1 0 -1 2 1 -2 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 1 -1 0 1 -1 0 1 -1 0 -2 2 0 -1 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 1 -1 -1 1 -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 -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 -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 -1] Vector space of degree 32 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 0 0 1 1 0 1 1 1/84 67/84 1/84 29/42 67/84 29/42 1/84 67/84 29/42 1/84 17/21 59/84 -4/21 1/2 67/84 125/84 -25/84 1/2 0 17/2 -1681/84 575/42] [ 0 0 1 -1 0 -1 1 0 0 0 1/84 67/84 -83/84 29/42 -17/84 -13/42 1/84 67/84 29/42 -83/84 -4/21 -25/84 -4/21 1/2 -17/84 41/84 -25/84 -1/2 0 15/2 -1681/84 533/42] [ 0 0 0 0 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 -16 41 -26] Vector space of degree 32 and dimension 1 over Rational Field Basis matrix: [ 0 0 0 0 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 -16 41 -26]
P = propeller_poset(4) P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 10 and dimension 7 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1] [ 0 0 1 0 0 0 0 0 0 -1] [ 0 0 0 1 0 0 0 0 0 -1] [ 0 0 0 0 1 0 1 0 0 3] [ 0 0 0 0 0 1 1 0 0 3] [ 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 10 and dimension 7 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1] [ 0 0 1 0 0 0 0 0 0 -1] [ 0 0 0 1 0 0 -1 0 0 0] [ 0 0 0 0 1 0 1 0 0 3] [ 0 0 0 0 0 1 1 0 0 3] [ 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 10 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1 0 0] [ 0 0 1 0 0 0 0 0 -1 0] [ 0 0 0 1 0 0 0 0 0 -1] [ 0 0 0 0 1 0 1 1 1 1] [ 0 0 0 0 0 1 1 1 1 1] Vector space of degree 10 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1 0 0] [ 0 0 1 0 0 0 0 0 -1 0] [ 0 0 0 1 0 0 -1 16 -41 25] [ 0 0 0 0 1 0 1 1 1 1] [ 0 0 0 0 0 1 1 1 1 1]
P = e6_minuscule_poset() P.show() antichain_space(P,1) order_ideal_space(P,1) #R.<q> = PolynomialRing(QQ) #antichain_space(P,q) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 27 and dimension 11 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 0 0 1 0 1 0 0 0 0 0 3 0 3 -1 -1 2 0 3 -2 1 3 1 0 0 4] [ 0 0 0 0 0 1 1 0 1 0 0 0 3 0 3 0 0 3 0 3 -1 2 3 2 0 0 6] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 2 2 0 2 0 0 4] [ 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 -1 1 1 0 0 -1 1 0 -1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 -1 0 1 -1 0 -1 2 1 -2 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 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 1 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 -1] Vector space of degree 27 and dimension 11 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 0 1 0 0 -1 0 -1 0 0 0 0 0 0 -1 -1 -1 0 0 -1 -1 0 -1 0 0 -2] [ 0 0 0 0 1 0 1 0 0 0 -1 0 3 -1 2 0 0 3 0 3 -2 1 3 1 0 0 4] [ 0 0 0 0 0 1 1 0 1 0 0 0 3 0 3 0 0 3 0 3 -1 2 3 2 0 0 6] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 2 2 0 2 0 0 4] [ 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 -1 1 0 -1 0 -1 2 1 -2 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 -1 0 1 -1 0 -2 2 0 -1 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 -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 1 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 -1] Vector space of degree 27 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 0 0 -1 -1 -1 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 -1 -1 -1 0 -1 -1 -1 0] [ 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 -1 -1 -1] [ 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 2 0 1 1 1 1 1 0] [ 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 2 2 1 2 1 2 1 2 2 1 1] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 2 1 1 1] Vector space of degree 27 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 1/57 1/57 41/57 41/57 0 40/57 1/57 0 1/57 67/57 0 22/19 0 0 22/19 -40/57 26/57 0 22/19 26/57 22/19 -210/19 569/19 -1000/57] [ 0 0 1 -1/57 -1/57 16/57 16/57 0 17/57 -1/57 0 -1/57 -10/57 0 -3/19 0 0 -3/19 -17/57 -26/57 0 -3/19 -26/57 -3/19 -94/19 210/19 -425/57] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 17 -40 27]
P = e7_minuscule_poset() P.show() antichain_space(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 56 and dimension 17 over Rational Field Basis matrix: 17 x 56 dense matrix over Rational Field Vector space of degree 56 and dimension 17 over Rational Field Basis matrix: 17 x 56 dense matrix over Rational Field Vector space of degree 56 and dimension 9 over Rational Field Basis matrix: 9 x 56 dense matrix over Rational Field Vector space of degree 56 and dimension 4 over Rational Field Basis matrix: 4 x 56 dense matrix over Rational Field
P = diagram_poset([2,1]).product(Posets.ChainPoset(3)) P.show() antichain_space(P,1) antichain_space_long(P,1) order_ideal_space(P,1) antichain_space(P,1).intersection(antichain_space(P,2)).intersection(antichain_space(P,13)) order_ideal_space(P,1).intersection(order_ideal_space(P,2)).intersection(order_ideal_space(P,13))
Vector space of degree 30 and dimension 2 over Rational Field Basis matrix: [ 0 1 0 0 2 2 0 0 2 0 2 2 0 0 2 2 2 0 2 1 1 1 1 1 1 1 1 1 1 1] [ 0 0 0 1 -1 -1 0 1 -1 1 0 0 1/2 1/2 -1/2 0 -1/2 1/2 -1/2 -1/2 1/2 -1/2 0 1/2 0 0 -1/2 1/2 0 0] [((0, 0), 0), ((0, 0), 1), ((0, 0), 2), ((0, 1), 0), ((0, 1), 1), ((0, 1), 2), ((1, 0), 0), ((1, 0), 1), ((1, 0), 2)] (1, 0, 0, 2, 1, 1, 0, 0, 0) (0, 0, 0, -1, -1/2, -1/2, 1, 1/2, 1/2) Vector space of degree 30 and dimension 2 over Rational Field Basis matrix: [ 0 1 1 0 0 0 1 0 0 0 -1 -1 1/2 1/2 -1/2 -1 -1/2 1/2 -1/2 1/2 -1/2 1/2 0 -1/2 0 0 1/2 -1/2 0 0] [ 0 0 0 1 -1 -1 0 1 -1 1 0 0 1/2 1/2 -1/2 0 -1/2 1/2 -1/2 -1/2 1/2 -1/2 0 1/2 0 0 -1/2 1/2 0 0] Vector space of degree 30 and dimension 0 over Rational Field Basis matrix: [] Vector space of degree 30 and dimension 0 over Rational Field Basis matrix: []