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## `SP::ToYY` -- converts any expression to the YY Schubert basis

#### Call(s)

`SP::ToYY(expr)`

#### Parameters

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

#### Introduction

The `SP::ToYY` function converts any expression expr to the YY Schubert basis. expr may involve some `x`i's, simple Schubert polynomials (`X[perm]`, `Y[code]`), double Schubert polynomials (`XX[perm]`, `YY[code]`, the second alphabet being the `y`i's), 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).

The call ``SP::ToYY`(expr, YY)` does not affect the argument `expr`, but it simplifies Schubert polynomials indices.

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 `ToYY(expr, b, Collect)`. For instance, ``SP::ToYY`(expr, YY, Collect)` may be used to collect the argument `expr`.

#### Example 1

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

                 5
          (q + 1)  (YY[0, 2, 0, 0] - y4 YY[0, 1, 0] +

             y3 YY[0, 0, 1, 0] + YY[0, 0, 1, 1, 0] - YY[0, 1, 0, 1, 0] +

             y3 (y4 YY[0] - YY[0, 0, 1, 0] + YY[0, 0, 0, 1, 0]))

    >> muEC::SP::ToYY(q^2*x3*X[3,1,2], Collect);

           2   2             2                2
          q  y1  y3 YY[0] - q  YY[2, 1, 0] - q  y1 YY[2, 0, 0] +

                        2          2           2
             YY[1, 0] (q  y1 y3 + q  y2 y3) + q  YY[2, 0, 1, 0] -

              2   2                2   2
             q  y1  YY[0, 1, 0] + q  y1  YY[0, 0, 1, 0] +

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

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

#### Related Functions

`ToXX`

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MuPAD Combinat, an open source algebraic combinatorics package