[Axiom-math] Re: [Axiom-developer] Examples for good types

[Martin Rubey]

Dear all,

Please reply only to axiom-math.

I very recently discovered another example, which I find (personally) more
convincing.

Axiom has a (rather weak, but still) domain for symmetric functions. 
The multiplication of two symmetric functions goes as follows:

(7) -> powerSum 4 * powerSum 2 * powerSum 2 * powerSum 2

            3
   (7)  (4 2 )
                                   Type: SymmetricPolynomial Fraction Integer

(i.e., it is represented as an integer partition, where equal parts are, as
customary, written as exponents.)

Symmetric Functions like determinants a lot, especially in Axiom:

m := matrix [[complete 1, complete 0],[complete 2, complete 1]]

        +     (1)        [] +
        |                   |
   (2)  |1       1   2      |
        |- (2) + - (1 )  (1)|
        +2       2          +
                            Type: Matrix SymmetricPolynomial Fraction Integer
(3) -> determinant m

          1       1   2
   (3)  - - (2) + - (1 )
          2       2
                                   Type: SymmetricPolynomial Fraction Integer

Note that Axiom uses the product in the ring of symmetric functions to compute
the determinant. To check, by Jacobi-Trudi the result should coincide with the
Schur function corresponding to the partition $(1,1)$:

(4) -> SFunction [1,1]

          1       1   2
   (4)  - - (2) + - (1 )
          2       2
                                   Type: SymmetricPolynomial Fraction Integer


Martin