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

 From: Martin Rubey Subject: Re: [Axiom-developer] Examples for good types Date: 28 Mar 2007 10:45:53 +0200 User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4

```Dear all,

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

```