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## `SP::TablePe` -- table of the Pe-basis

#### Call(s)

`SP::TablePe(n)`

#### Parameters

`n` | - | any positive integer denoting the degree of a symmetric group

#### Introduction

The `SP::TablePe` function returns the table of all elements of the `Pe`-basis, which is the dual basis of the basis of (increasing) 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)`.

When called with the second argument `X`, it returns the table of all polynomials in Sn, expressed in the Schubert basis.

#### Example 1

    >> t:=muEC::SP::TablePe(3);

                            table(
                                            2
                              [2, 1, 0] = x1  x2,
                                                    2
                              [1, 1, 0] = x1 x2 + x1 ,
                              [1, 0, 0] = x1 + x2,
                              [2, 0, 0] = x1 x2,
                              [0, 1, 0] = x1,
                              [0, 0, 0] = 1
                            )

    >> muEC::SP::TablePe(3, X);

                      table(
                        [2, 1, 0] = X[3, 2, 1],
                        [1, 1, 0] = X[2, 3, 1] + X[3, 1, 2],
                        [1, 0, 0] = X[1, 3, 2],
                        [2, 0, 0] = X[2, 3, 1],
                        [0, 1, 0] = X[2, 1],
                        [0, 0, 0] = X[1]
                      )

#### Related Functions

`TableX`, `TableXX`

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MuPAD Combinat, an open source algebraic combinatorics package