def canonical_scheme(self, t=None): """ Return the canonical scheme. This is a scheme that contains this hypergeometric motive in its cohomology. EXAMPLES:: sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp sage: H = Hyp(cyclotomic=([3],[4])) sage: H.gamma_list() [-1, 2, 3, -4] sage: H.canonical_scheme() Spectrum of Quotient of Multivariate Polynomial Ring in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field by the ideal (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^2*X1^3 + 27/64*Y0*Y1^4) sage: H = Hyp(gamma_list=[-2, 3, 4, -5]) sage: H.canonical_scheme() Spectrum of Quotient of Multivariate Polynomial Ring in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field by the ideal (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^3*X1^4 + 1728/3125*Y0^2*Y1^5) REFERENCES: [Kat1991]_, section 5.4 """ if t is None: t = FractionField(QQ['t']).gen() basering = t.parent() gamma_pos = [u for u in self.gamma_list() if u > 0] gamma_neg = [u for u in self.gamma_list() if u < 0] N_pos = len(gamma_pos) N_neg = len(gamma_neg) varX = ['X{}'.format(i) for i in range(N_pos)] varY = ['Y{}'.format(i) for i in range(N_neg)] ring = PolynomialRing(basering, varX + varY) gens = ring.gens() X = gens[:N_pos] Y = gens[N_pos:] eq0 = ring.sum(X) - 1 eq1 = ring.sum(Y) - 1 eq2_pos = ring.prod(X[i] ** gamma_pos[i] for i in range(N_pos)) eq2_neg = ring.prod(Y[j] ** -gamma_neg[j] for j in range(N_neg)) ideal = ring.ideal([eq0, eq1, self.M_value() * eq2_neg - t * eq2_pos]) return Spec(ring.quotient(ideal))
def canonical_scheme(self, t=None): """ Return the canonical scheme. This is a scheme that contains this hypergeometric motive in its cohomology. EXAMPLES:: sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp sage: H = Hyp(cyclotomic=([3],[4])) sage: H.gamma_list() [-1, 2, 3, -4] sage: H.canonical_scheme() Spectrum of Quotient of Multivariate Polynomial Ring in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field by the ideal (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^2*X1^3 + 27/64*Y0*Y1^4) sage: H = Hyp(gamma_list=[-2, 3, 4, -5]) sage: H.canonical_scheme() Spectrum of Quotient of Multivariate Polynomial Ring in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field by the ideal (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^3*X1^4 + 1728/3125*Y0^2*Y1^5) REFERENCES: [Kat1991]_, section 5.4 """ if t is None: t = FractionField(QQ['t']).gen() basering = t.parent() gamma_pos = [u for u in self.gamma_list() if u > 0] gamma_neg = [u for u in self.gamma_list() if u < 0] N_pos = len(gamma_pos) N_neg = len(gamma_neg) varX = ['X{}'.format(i) for i in range(N_pos)] varY = ['Y{}'.format(i) for i in range(N_neg)] ring = PolynomialRing(basering, varX + varY) gens = ring.gens() X = gens[:N_pos] Y = gens[N_pos:] eq0 = ring.sum(X) - 1 eq1 = ring.sum(Y) - 1 eq2_pos = ring.prod(X[i] ** gamma_pos[i] for i in range(N_pos)) eq2_neg = ring.prod(Y[j] ** -gamma_neg[j] for j in range(N_neg)) ideal = ring.ideal([eq0, eq1, self.M_value() * eq2_neg - t * eq2_pos]) return Spec(ring.quotient(ideal))
def expand(self, n, letter='x'): r""" Expand ``self`` written in the `\mathbf{w}` basis in `n^2` commuting variables which satisfy the relation `x_{ij} x_{ik} = 0` for all `i`, `j`, and `k`. The expansion of an element of the `\mathbf{w}` basis is given by equations (26) and (55) in [HNT06]_. INPUT: - ``n`` -- an integer - ``letter`` -- (default: ``'x'``) a string OUTPUT: - The symmetric function of ``self`` expressed in the ``n*n`` non-commuting variables described by ``letter``. REFERENCES: .. [HNT06] \F. Hivert, J.-C. Novelli, J.-Y. Thibon. *Commutative combinatorial Hopf algebras*. (2006). :arxiv:`0605262v1`. EXAMPLES:: sage: w = SymmetricFunctionsNonCommutingVariables(QQ).dual().w() sage: w[[1,3],[2]].expand(4) x02*x11*x20 + x03*x11*x30 + x03*x22*x30 + x13*x22*x31 One can use a different set of variable by using the optional argument ``letter``:: sage: w[[1,3],[2]].expand(3, letter='y') y02*y11*y20 """ from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.combinat.permutation import Permutations m = self.parent() names = ['{}{}{}'.format(letter, i, j) for i in range(n) for j in range(n)] R = PolynomialRing(m.base_ring(), n*n, names) x = [[R.gens()[i*n+j] for j in range(n)] for i in range(n)] I = R.ideal([x[i][j]*x[i][k] for j in range(n) for k in range(n) for i in range(n)]) Q = R.quotient(I, names) x = [[Q.gens()[i*n+j] for j in range(n)] for i in range(n)] P = SetPartitions() def on_basis(A): k = A.size() ret = R.zero() if n < k: return ret for p in Permutations(k): if P(p.to_cycles()) == A: # -1 for indexing ret += R.sum(prod(x[I[i]][I[p[i]-1]] for i in range(k)) for I in Subsets(range(n), k)) return ret return m._apply_module_morphism(self, on_basis, codomain=R)