gaugeMatrix -- computes the base change over the field of rational functions

Usage:

gaugeMatrix(G, B)
Inputs:
- I, an ideal, of the Weyl algebra
- G, a list, of generators of a Gröbner basis for $I$
- B, a list, of standard monomials for $I$, i.e. any basis of $R_n/R_nI$ over the field of rational functions
Outputs:
- M, a matrix, encoding the change of basis with regard to to $B$

Description

Let $I = D_n\langle G\rangle$ be a $D_n$-ideal, $B$ basis of $R_n/R_nI$. This methods computes the matrix which encodes the change of basis from the set of standard monomials of $R_nI$ to the basis $B$.

i1 : D1 = makeWeylAlgebra(QQ[x, y], w1 = {2, 1});

i2 : I = ideal(x*dx^2-y*dy^2+2*dx-2*dy, x*dx+y*dy+1);

o2 : Ideal of D1

i3 : SM1 = standardMonomials I

o3 = {1, dy}

o3 : List

i4 : F = baseFractionField D1;

i5 : D2 = makeWeylAlgebra(QQ[x, y], w2 = {1, 2});

i6 : SM2 = standardMonomials sub(I, D2)

o6 = {1, dx}

o6 : List

i7 : gaugeMatrix(I, SM2)

o7 = | 1    0    |
     | -1/x -y/x |

             2      2
o7 : Matrix F  <-- F

It is also possible to compute the gauge matrix of a system of connection matrices containing parameters.

i8 : D = makeWeylAlgebra(frac(QQ[a])[x]);

i9 : I = ideal(x^2*dx^2 + x*dx + (x^2-a^2))

            2  2           2    2
o9 = ideal(x dx  + x*dx + x  - a )

o9 : Ideal of D

i10 : standardMonomials I

o10 = {1, dx}

o10 : List

i11 : gaugeMatrix(I,{dx,x^2*dx^2+x*dx+x^2})

o11 = | 0  1 |
      | a2 0 |

                             2                     2
o11 : Matrix (frac(QQ[x, a]))  <-- (frac(QQ[x, a]))

See Cosmological correlator for the 2-site chain for an example.

Ways to use gaugeMatrix:

gaugeMatrix(Ideal,List)
gaugeMatrix(List,List)

For the programmer

The object gaugeMatrix is a method function.

The source of this document is in ConnectionMatrices/docs.m2:177:0.

gaugeMatrix -- computes the base change over the field of rational functions

Description

See also

Ways to use gaugeMatrix:

For the programmer