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## `SP::ToPe` -- converts any expression to the Pe-basis

#### Call(s)

`SP::ToPe(expr)`

#### Parameters

`expr` | - | any expression
`b` | - | any name of a known basis

#### Introduction

The `SP::ToPe` function converts any expression expr to the `Pe`-basis, which is the dual basis of the basis of monomials less than the ``staircase'' monomial, i.e. the basis of products of elementary symmetric functions on an alphabet flag. For instance, `Pe[1, 2, 0, 1, 0]` stands for the product `e1(x1,x2,x3,x4)*e2(x1,x2,x3)*e1(x1)`.

`expr` may involve some `x`i's, simple Schubert polynomials (`X[perm]`, `Y[code]`), double Schubert polynomials (`XX[perm]`, `YY[code]`), product of elementary functions (`Pe[vect]`), other terms being considered as coefficients.

The expression `expr` is expanded and the result is not collected.

One may specify by a second argument, say `b`, that `expr` is solely expressed in terms of the known basis `b` (`x`, `X`, `Y`, `XX`, `YY`, `Pe` and even `y` that is seen as a basis in the package).

One may add `NoExpand` just after the argument `expr` to choose not to expand the expression expr before treating it.

One may collect the result by adding a third argument: this is done by `ToPe(expr, b, Collect)`.

#### Example 1

    >> muEC::SP::ToPe((1+q)^5*x3*x4, NoExpand);

          (Pe[1, 0, 0] - Pe[1, 0, 0, 0]) (Pe[1, 0, 0, 0] -

                                       5
             Pe[1, 0, 0, 0, 0]) (q + 1)

    >> muEC::SP::ToPe(q^2*x3*XX[3,1,2]*Y[0,1], Collect);

            2       2                     2       2
          (q  y1 + q  y2) Pe[2, 1, 0] - (q  y1 + q  y2) Pe[2, 0, 1, 0] -

              2                   2                   2
             q  Pe[1, 2, 1, 0] + q  Pe[2, 1, 1, 0] + q  Pe[3, 0, 1, 0] -

              2                   2
             q  Pe[3, 1, 0, 0] - q  y1 y2 Pe[2, 0, 0] +

              2
             q  y1 y2 Pe[2, 0, 0, 0]

#### Related Functions

`Tox`, `ToX`, `ToXX`

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MuPAD Combinat, an open source algebraic combinatorics package