UsePrimePowers -- use a diagonal weight matrix with multiples of certain powers

Synopsis

Usage:

Lgr = matchingFieldFromPermutation(k, n, S, UsePrimePowers => b, PowerValue => v)

Lfl = matchingFieldFromPermutation(kList, n, S, UsePrimePowers => b, PowerValue => v)
Inputs:
- k, an integer,
- kList, a list,
- n, an integer,
- S, a list, a permutation of $1, \dots, n$
- r, a Boolean value, set whether prime powers are to be used
- v, an integer, set the value of the powers to be used
Outputs:
- Lgr, an instance of the type GrMatchingField,
- Lfl, an instance of the type FlMatchingField,

Description

The options UsePrimePowers and PowerValue are optional arguments for the function matchingFieldFromPermutation, which constructs a matching field by permuting a row (usually the second row) of a weight matrix that gives rise to the diagonal matching field.

If the option UsePrimePowers is set to true, then the underlying diagonal weight matrix is given by $M_0 = (m_{i,j})$ with $m_{i,j} = (n-j) p^(i-2)$ if $i > 1$ and $m_{i,j} = 0$ if $i = 1$, where $p \ge n$ is the smallest prime number greater than or equal to $n$.

i1 : getWeightMatrix matchingFieldFromPermutation(3, 6, {1,2,3,4,5,6}, UsePrimePowers => false)

o1 = | 0  0  0  0  0 0 |
     | 1  2  3  4  5 6 |
     | 30 24 18 12 6 0 |

              3       6
o1 : Matrix ZZ  <-- ZZ

i2 : getWeightMatrix matchingFieldFromPermutation(3, 6, {1,2,3,4,5,6}, UsePrimePowers => true)

o2 = | 0  0  0  0  0 0 |
     | 1  2  3  4  5 6 |
     | 35 28 21 14 7 0 |

              3       6
o2 : Matrix ZZ  <-- ZZ

To set the value of $p$ in the above matrix to a specific value, use the option PowerValue.

i3 : getWeightMatrix matchingFieldFromPermutation(3, 6, {1,2,3,4,5,6}, PowerValue => 10)

o3 = | 0  0  0  0  0  0 |
     | 1  2  3  4  5  6 |
     | 50 40 30 20 10 0 |

              3       6
o3 : Matrix ZZ  <-- ZZ

The option PowerValue overrides the option UsePrimePowers.

Warning. Certain values for the option PowerValue will produce weight matrices that are not generic, i.e., the initial term of the corresponding Pluecker forms are not all monomials, hence the weight matrix does not define a matching field. The function matchingFieldFromPermutation will produce a matching field, which may lead to unexpected behaviours.

Precise details. The value of $p$ is determined as follows. The option PowerValue is used if it is a positive value. If PowerValue is not positive then the function checks to see if the option UsePrimePowers is set to true. If it is, then the value of $p$ is set to be the small prime number greater than or equal to $n$. Otherwise, if UsePrimePowers is set to false, then $p$ is set to be $n$.

Functions with optional argument named UsePrimePowers :

matchingFieldFromPermutation(...,UsePrimePowers=>...) -- see matchingFieldFromPermutation -- matching field parametrised by permutations

For the programmer

The object UsePrimePowers is a symbol.

UsePrimePowers -- use a diagonal weight matrix with multiples of certain powers

Synopsis

Description

See also

Functions with optional argument named UsePrimePowers :

For the programmer

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