1 | initial version |

This is a bit more subtle. Here is a way.

```
sage: P.<x,y>=PolynomialRing(QQ['a','b'])
sage: BR = P.base_ring()
sage: S1 = SymmetricFunctions(QQ)
sage: S2 = SymmetricFunctions(BR)
sage: a,b = BR.gens()
sage: f = x+y+a+b
sage: step1 = S2.from_polynomial(f);step1
(a+b)*m[] + m[1]
sage: sum(S1.m()(i).tensor(S1.from_polynomial(c)) for i,c in step1)
m[] # m[1] + m[1] # m[]
```

The key point to understand is that you need to explain sage in what you answer lives. Here it is the tensor product of symmetric functions with themselves.

2 | No.2 Revision |

This is a bit more subtle. Here is a way.

```
sage: P.<x,y>=PolynomialRing(QQ['a','b'])
sage: BR = P.base_ring()
sage: S1 = SymmetricFunctions(QQ)
sage: S2 = SymmetricFunctions(BR)
sage: a,b = BR.gens()
sage: f = x+y+a+b
sage: step1 = S2.from_polynomial(f);step1
(a+b)*m[] + m[1]
sage: sum(S1.m()(i).tensor(S1.from_polynomial(c)) for i,c in step1)
m[] # m[1] + m[1] # m[]
```

The key point to understand is that you need to explain sage in what ~~you ~~your answer lives. Here it is the tensor product of symmetric functions with themselves.

3 | No.3 Revision |

This is a bit more subtle. Here is a way.

```
sage: P.<x,y>=PolynomialRing(QQ['a','b'])
sage: BR = P.base_ring()
sage: S1 =
```~~SymmetricFunctions(QQ)
~~SymmetricFunctions(QQ).e()
sage: S2 = ~~SymmetricFunctions(BR)
~~SymmetricFunctions(BR).e()
sage: a,b = BR.gens()
sage: f = x+y+a+b
sage: step1 = S2.from_polynomial(f);step1
~~(a+b)*m[] ~~(a+b)*e[] + ~~m[1]
~~e[1]
sage: ~~sum(S1.m()(i).tensor(S1.from_polynomial(c)) ~~sum(S1(i).tensor(S1.from_polynomial(c)) for ~~i,c ~~i, c in step1)
~~m[] ~~e[] # ~~m[1] ~~e[1] + ~~m[1] ~~e[1] # ~~m[]
~~e[]

The key point to understand is that you need to explain sage in what your answer lives. Here it is the tensor product of symmetric functions with themselves.

EDIT: I have changed the answer above to use the elementary basis.

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.