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# Symmetric functions defined by orthogonality and triangularity.

One characterization of Schur functions is that they are upper triangularly related to the monomial symmetric functions and orthogonal with respect to the Hall scalar product. We can use the class SymmetricFunctionAlgebra_orthotriang to obtain the Schur functions from this definition.

    sage: from sage.combinat.sf.sfa import zee
    sage: from sage.combinat.sf.orthotriang import SymmetricFunctionAlgebra_orthotriang
    sage: Sym = SymmetricFunctions(QQ)
    sage: m = Sym.m()
    sage: s = SymmetricFunctionAlgebra_orthotriang(Sym, m, zee, 's', 'Schur functions')
    sage: s([2,1])^2
    s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]

    sage: s2 = SymmetricFunctions(QQ).s()
    sage: s2([2,1])^2
    s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]

class `sage.combinat.sf.orthotriang.``SymmetricFunctionAlgebra_orthotriang`(Sym, base, scalar, prefix, basis_name, leading_coeff=None)

  Bases: `sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic`

  Initialization of the symmetric function algebra defined via orthotriangular rules.

  INPUT:

  - `self` – a basis determined by an orthotriangular definition
  - `Sym` – ring of symmetric functions
  - `base` – an instance of a basis of the ring of symmetric functions (e.g. the Schur functions)
  - `scalar` – a function `zee` on partitions. The function `zee` determines the scalar product on the power sum basis with normalization \(\langle p_{\mu}, p_{\mu} \rangle = \mathrm{zee}(\mu)\).
  - `prefix` – the prefix used to display the basis
  - `basis_name` – the name used for the basis

  Note

  The base ring is required to be a \(\QQ\)-algebra for this method to be useable, since the scalar product is defined by its values on the power sum basis.

  EXAMPLES:

      sage: from sage.combinat.sf.sfa import zee
      sage: from sage.combinat.sf.orthotriang import SymmetricFunctionAlgebra_orthotriang
      sage: Sym = SymmetricFunctions(QQ)
      sage: m = Sym.m()
      sage: s = SymmetricFunctionAlgebra_orthotriang(Sym, m, zee, 's', 'Schur'); s
      Symmetric Functions over Rational Field in the Schur basis

  TESTS:

      sage: TestSuite(s).run(elements = [s[1,1]+2*s[2], s[1]+3*s[1,1]])
      sage: TestSuite(s).run(skip = ["_test_associativity", "_test_prod"])  # long time (7s on sage.math, 2011)

  Note: `s.an_element()` is of degree 4; so we skip `_test_associativity` and `_test_prod` which involve (currently?) expensive calculations up to degree 12.

  class `Element`(M, x)

    Bases: `sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element`

    Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s `__call__()` method.

    TESTS:

        sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
        sage: B = F.basis()
        sage: f = B['a'] + 3*B['c']; f
        B['a'] + 3*B['c']
        sage: f == loads(dumps(f))
        True

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