cayleypy.MatrixGroups

class cayleypy.MatrixGroups[source]

Pre-defined Cayley graphs for matrix groups.

__init__()

Methods

__init__()

heisenberg(*[, n, modulo, add_inverses])

Returns Cayley graph for the Heisenberg group.

special_linear_fundamental_roots(n[, modulo])

Returns Cayley graph for the special linear group SL(n, Z/mZ).

special_linear_root_weyl(n[, modulo])

Returns Cayley graph for the special linear group SL(n, Z/mZ).
static heisenberg(*, n: int = 3, modulo: int = 0, add_inverses: bool = True) → CayleyGraphDef[source]

Returns Cayley graph for the Heisenberg group.

For n=3, this is a group of upper triangular 3x3 integer matrices with 1s on main diagonal. See https://en.wikipedia.org/wiki/Heisenberg_group. Generated by 4 matrices: x=(110,010,001), y=(100,011,001), and their inverses.

For n>=4, this is a group of n*n matrices which differ from identity matrix only in top row or right column. See https://en.wikipedia.org/wiki/Heisenberg_group#Higher_dimensions.

The number of generators is 4(n-2) when inverses are added or 2(n-2) when inverses are not added.

Central element is identity matrix.

Parameters:

  • n – Size of the matrix. Defaults to 3.

  • modulo – multiplication modulo (or 0 if multiplication is not modular). Defaults to 0.

  • add_inverses – whether to add inverse generators. Defaults to True.

Returns:

requested graph as CayleyGraphDef.

static special_linear_fundamental_roots(n: int, modulo: int = 0) → CayleyGraphDef[source]

Returns Cayley graph for the special linear group SL(n, Z/mZ).

This is a group of n x n integer matrices with determinant 1, modulo m.

Generated by n-1 fundamental roots: e_i = (e_ij) where e_ij is 1 at (i,j) and 0 elsewhere, and their negatives. Central element is identity matrix.

Parameters:

  • n – Size of matrices.

  • modulo – multiplication modulo (or 0 if multiplication is not modular).

Returns:

requested graph as CayleyGraphDef.

static special_linear_root_weyl(n: int, modulo: int = 0) → CayleyGraphDef[source]

Returns Cayley graph for the special linear group SL(n, Z/mZ).

This is a group of n x n integer matrices with determinant 1, modulo m.

Generated by a single root element e_12 and a lift of a Coxeter element w. Central element is identity matrix.

Parameters:

  • n – Size of matrices.

  • modulo – multiplication modulo (or 0 if multiplication is not modular).

Returns:

requested graph as CayleyGraphDef.