SP::ToPe --
converts any expression to the Pe-basis
Call(s)
SP::ToPe(expr)
Parametersexpr | - | 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 xi'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 FunctionsMuPAD Combinat, an open source algebraic combinatorics package