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toBracketPolynomial -- Represent an invariant polynomial as a polynomial in brackets

Description

A polynomial invariant under the action of the special linear group may be represented by a polynomial in brackets. See Bracket for more details. Groebner basis methods allow one to compute such a representation. Below is a simple example.

i1 : B = bracketRing(3, 2)

o1 = B
      3,2

o1 : BracketRing
i2 : X = matrix B

o2 = {-1} | x_(1,1) x_(1,2) |
     {-1} | x_(2,1) x_(2,2) |
     {-1} | x_(3,1) x_(3,2) |

                                                 3                                          2
o2 : Matrix (QQ[x   ..x   , y    , y    , y    ])  <-- (QQ[x   ..x   , y    , y    , y    ])
                 1,1   3,2   [23]   [13]   [12]             1,1   3,2   [23]   [13]   [12]
i3 : f = (det X^{1,2}) - (det X^{0,2}) + (det X^{0,1})

o3 = - x   x    + x   x    + x   x    - x   x    - x   x    + x   x
        1,2 2,1    1,1 2,2    1,2 3,1    2,2 3,1    1,1 3,2    2,1 3,2

o3 : QQ[x   ..x   , y    , y    , y    ]
         1,1   3,2   [23]   [13]   [12]
i4 : toBracketPolynomial(f,B)

o4 = [23]-[13]+[12]

o4 : GCExpression

Such a representation is not unique. It may be checked that two bracket polynomials are equal through their normal form with respect to a Groebner basis. See normalForm for a further explanation.

See also BracketRing.

Ways to use toBracketPolynomial:

  • toBracketPolynomial(RingElement,BracketRing)

For the programmer

The object toBracketPolynomial is a method function.


The source of this document is in Brackets.m2:547:0.