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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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