Database Cubic Hecke Algebras

This repository contains data for the representations of the Cubic Hecke Algebra calculated by Ivan Marin. The original data of Ivan Marin are published in a format which can be read by Maple. The purpose of this repository is, to make them available in a Python like style such that they can be easily installed into SageMath using pip.

This repository was created as a part of the SageMath functionality for the cubic Hecke algebras (see Trac ticket #29717)

In addition to Ivan Marin's data it contains coefficients for linear forms on the cubic Hecke algebras on up to four strands satisfying the Markov trace condition (see for example Louis Kauffman: Knots and Physics, sections 7.1 and 7.2). This data has been precomputed with the SageMath functionality introduced by the above mentioned ticket (see Python module create_markov_trace_data.py).

Usage

In Python, it can be used as follows:

>>> from database_cubic_hecke import read_basis
>>> b4 = read_basis()
>>> len(b4)
648
>>> b2 = read_basis(num_strands=2); b2
[[], [1], [-1]]
>>> b3 = read_basis(num_strands=3)
>>> len(b3)
24

>>> from database_cubic_hecke import read_irr
>>> dim_list, repr_list, repr_list_inv = read_irr()
>>> dim_list
[1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 8, 8, 8, 9, 9]
>>> repr_list[5][1]
{(0, 0): c, (0, 1): -1, (1, 1): a}
>>> from math import sqrt
>>> j = (sqrt(3)*1j-1)/2
>>> dim_list, repr_list, repr_list_inv = read_irr((5, 7, 3, j))
>>> repr_list[23][0][(3, 8)]
(1.5+6.06217782649107j)

>>> from database_cubic_hecke import read_reg
>>> dim_list, repr_list, repr_list_inv = read_reg()
>>> dim_list
[648]
>>> [len(m) for m in repr_list[0]]
[1080, 1701, 7862]
>>> [len(m) for m in repr_list_inv[0]]
[1080, 1728, 9370]
>>> dim_list, repr_list, repr_list_inv = read_reg(num_strands=3)
>>> dim_list
[24]
>>> [len(m) for m in repr_list[0]]
[40, 63]

>>> from database_cubic_hecke.markov_trace_coeffs import read_markov
>>> read_markov('U2', (3,5,7,11), num_strands=3)
[0, 11, 0.09090909090909091, 11, 0.09090909090909091, 0, 0, 0, 0, -55, 11, 11,
 -4.714285714285714, -0.45454545454545453, 0.09090909090909091, 0, 0, 0, 0,
 0.09090909090909091, -0.03896103896103896, -0.45454545454545453, 0, 0]

If you have SymPy installed you can obtain representation matrices directly:

>>> from database_cubic_hecke import irr_reprs_matrices
>>> m1, m2, m3 = irr_reprs_matrices(5)
>>> m1i, m2i, m3i = irr_reprs_matrices(5, inverse=True)
>>> m1 * m1i
Matrix([
[1, 0],
[0, 1]])
>>> m1*m2*m1 == m2*m1*m2
True
>>> m1i*m2i*m1i == m2i*m1i*m2i
True

>>> from database_cubic_hecke import reg_reprs_matrices
>>> m1, m2, m3 = reg_reprs_matrices()
>>> m1.shape
(648, 648)
>>> m1i, m2i = reg_reprs_matrices(inverse=True, num_strands=3)
>>> m1i.shape
(24, 24)
>>> m1i*m2i*m1i == m2i*m1i*m2i
True

>>> from database_cubic_hecke.markov_trace_coeffs import read_markov
>>> from sympy import var
>>> u, v, w, s = var('u, v, w, s')
>>> variables = (u, v, w, s)
>>> read_markov('U2', variables, num_strands=3)
[0, s, 1/s, s, 1/s, 0, 0, 0, 0, -s*v, s, s, -s*u/w, -v/s, 1/s,
0, 0, 0, 0, 1/s, -u/(s*w), -v/s, 0, 0]

The usage in Sage will be implicitely via the new class CubicHeckeAlgebra according to the Trac ticket #29717. But anyway, it can also be used indenpendently, for example:

sage: from database_cubic_hecke import read_irr
sage: F = CyclotomicField(3)
sage: L.<a, b, c> = LaurentPolynomialRing(F)
sage: T = L.gens_dict_recursive()
sage: T['j'] = T['zeta3']
sage: T.pop('zeta3')
sage: irr = read_irr(tuple(T.values()))
sage: dim_list, repr_list, repr_list_inv= irr
sage: m1d, m2d , m3d = repr_list[23]
sage: d = dim_list[23]
sage: m1 = matrix(d, d, m1d)
sage: m2 = matrix(d, d, m2d)
sage: m3 = matrix(d, d, m3d)
sage: m1
[             c              0              0              0              0              0              0              0              0]
[     b^2 + a*c              b              0              0              0              0   (-zeta3)*b*c              0              0]
[             b              1              a              0              0              0              c              0              0]
[             0              0              0              a              0              0             -c (-zeta3 - 1)*c    a + zeta3*b]
[   zeta3*a - b              0              0              0              b              0              0              0              0]
[       zeta3*a              0              0              0              b              a              0              0              0]
[             0              0              0              0              0              0              c              0              0]
[             0              0              0              0              0              0              0              c              0]
[             0              0              0              0              0              0              0        zeta3*c              b]

sage: m1*m2*m1 == m2*m1*m2
True
sage: m3*m2*m3 == m2*m3*m2
True
sage: m3*m1 == m1*m3
True


sage: from database_cubic_hecke import read_reg
sage: R.<u, v, w> = ZZ[]
sage: B = R.localization(w)
sage: T = B.gens_dict_recursive()
sage: reg = read_reg(tuple(T.values()))
sage: dim_list, repr_list, repr_list_inv= reg
sage: m1d, m2d , m3d = repr_list[0]
sage: d = dim_list[0]
sage: m1 = matrix(d, d, m1d)
sage: m2 = matrix(d, d, m2d)
sage: m3 = matrix(d, d, m3d)
sage: m1
648 x 648 sparse matrix over Multivariate Polynomial Ring in u, v, w over Integer Ring localized at (w,) (use the '.str()' method to see the entries)

sage: m1*m2*m1 == m2*m1*m2
True
sage: m3*m2*m3 == m2*m3*m2
True
sage: m3*m1 == m1*m3
True

To build a new release, the files containing the data in Python syntax can be upgraded with the create_marin_data script. There is a workflow to run this script and build a new release if differences are detected. It can be triggered manually.

Installation

Python

pip install database_cubic_hecke

or

pip install database_cubic_hecke==2022.3.5

if you want to install a former version.

SageMath

After release of the above mentioned Trac ticket, the database can be installed in Sage by:

sage -i database_cubic_hecke

This will contain integration with the cubic Hecke algebra functionality of Sage. Before, or to use it independent on the new Sage functionality the installation works as follows:

sage -pip install database_cubic_hecke

or

sage -pip install database_cubic_hecke==2022.3.5

for a special version.

Versioning

Version numbers are automatically generated on a manually triggered workflow Check version changed if differences to the original databases are detected. They follow the scheme

<year>.<month>.<day>

with respect to the date the workflow is triggered.

Help

If you note a divergence between this repository and the original data in case the current release is older than a month please create an issue about that.

Credits

Many thanks to Ivan Marin to make his data available for their use in Sage.