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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
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: []