Constructing a dual pair of algebras from a table¶
Constructing a dual pair of algebras from a table.
- dual_pairs.dual_pair_from_table.algebra_and_points_from_action(G, V, action, reduced=True)¶
This is a helper function for
dual_pair_from_table()
.EXAMPLES:
sage: from dual_pairs.dual_pair_from_table import algebra_and_points_from_action sage: R.<x> = QQ[] sage: L.<a> = NumberField(x^3 - x^2 - 2*x + 1) sage: G = L.galois_group() sage: V = GF(2)^2 sage: M = MatrixSpace(GF(2), 2, 2) sage: table = {G[0]: M.one(), ....: G[1]: M([1, 1, 1, 0]), ....: G[2]: M([0, 1, 1, 1])} sage: algebra_and_points_from_action(G, V, lambda g, v: table[g] * v) ( Finite flat algebra of degree 4 over Rational Field, product of: Number Field in a0 with defining polynomial x Number Field in a1 with defining polynomial x^3 - x^2 - 2*x + 1 , [(0, 0), (0, 1), (1, 1), (1, 0)], [ 1 0 0 0] [ 0 1 -a^2 + 2 -a^2 + a + 3] [ 0 1 a^2 - a - 1 -a + 2] [ 0 1 a a^2] )
- dual_pairs.dual_pair_from_table.dual_pair_from_table(G, V, table, reduced=True)¶
Return a dual pair of algebras corresponding to the given Galois representation.
INPUT:
G – Galois group of a finite Galois extension of \(\mathbf{Q}\)
V – a finite-dimensional vector space over a finite field
table
– a dictionary{g: rho(g)}
where g ranges over G andrho
is a group homomorphism from G to the automorphism group of V.reduced
– boolean (default:True
); whether to applypolredabs
to the defining polynomials of number fields appearing in the output
EXAMPLES:
sage: from dual_pairs.dual_pair_from_table import dual_pair_from_table sage: R.<x> = QQ[] sage: L.<a> = NumberField(x^3 - x^2 - 2*x + 1) sage: G = L.galois_group() sage: V = GF(2)^2 sage: M = MatrixSpace(GF(2), 2, 2) sage: table = {G[0]: M.one(), ....: G[1]: M([1, 1, 1, 0]), ....: G[2]: M([0, 1, 1, 1])} sage: dual_pair_from_table(G, V, table) Dual pair of algebras over Rational Field A = Finite flat algebra of degree 4 over Rational Field, product of: Number Field in a0 with defining polynomial x Number Field in a1 with defining polynomial x^3 - x^2 - 2*x + 1 B = Finite flat algebra of degree 4 over Rational Field, product of: Number Field in a0 with defining polynomial x Number Field in a1 with defining polynomial x^3 - x^2 - 2*x + 1