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.
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 (see Python module create_markov_trace_data.py).
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 is supported by the class CubicHeckeAlgebra which is available since version 9.7. If you don't have Sage or an older version on your own device you may use the functionality online:
There is a tutorial (also available as pdf) about the Sage class which can also be used in these cloud applications.
The use of the cubic Hecke algebra on less than four strands is also possible on other Sage cloud installations.
Anyway, you can also use this Python wrapper in older Sage versions, 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
Truepip install database_cubic_heckeor
pip install database_cubic_hecke==2022.3.5if you want to install a former version.
If you have a Sage release 9.7 or newer the database can be installed by:
sage -i database_cubic_heckeThis will contain integration with the cubic Hecke algebra functionality of Sage.
If your Sage version is older than 9.7 or if you have problems with the above
installation method you can use pip instead:
sage -pip install database_cubic_heckeor
sage -pip install database_cubic_hecke==2022.3.5for a special version.
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.
To build a new release manually, the files containing the data in Python syntax can be upgraded with the create_marin_data script script. This is performed by the above mentioned workflow.
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.
Many thanks to Ivan Marin to make his data available for their use in Sage.