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class DiagonalPolynomialRing:
    """
    The ring of diagonal polynomials in n x r variables + n x inert variables

    EXAMPLES::
   
        sage: DiagonalPolynomialRing(QQ, n, r, inert=0)

    """
    def young_idempotent(self, p, mu):
    def polarization(self, p, ...):
    ...

class DiagonalAntisymmetricPolynomials:
    """
    The subspace of diagonal polynomials that are antisymmetric w.r.t. certain variables

    Polynomials are represented by picking within each orbit of monomials a canonical one.
    """
    def from_polynomial(self, p):
          '''Create an element from a fully expanded antisymmetric polynomial'''
    def young_idempotent(self, p):
         ''''''
    def polarization()
          '''compute usual polarization and renormalize'''

def PolarizationSpace(P, generators, mu, r, use_symmetry=, verbose=, use_lie, use_commutativity???=):
    """
    Starting from  polynomials in the mu-isotypic component of the polynomial ring in one set of variables (possibly with additional inert variables), construct the space obtained by polarization.

    P: a diagonal polynomial ring (or assymmetric version)
    generators: polynomials in one set of variables (+inert) in the image of b_mu
    """

def harmonic_XXX(mu, n):
    '''construct a basis of the mu-isotypic component of harmonic polynomials in n variables through higher specht polynomials'''

def YYY(mu, nu, n):
    '''The analogue for Pauline's case'''