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Trac #34412: q-commuting polynomials
These show up in quantum mechanics computations and are a variation of
skew commuting polynomials (cf. the exterior algebra) satisfying `yx =
q^{B_xy} xy` for all variables that index fixed a skew symmetric
bilinear matrix `B`.
We provide a simple initial implementation.
URL: https://trac.sagemath.org/34412
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): Frédéric Chapoton- Loading branch information
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,351 @@ | ||
| r""" | ||
| `q`-Commuting Polynomials | ||
| AUTHORS: | ||
| - Travis Scrimshaw (2022-08-23): Initial version | ||
| """ | ||
|
|
||
| # **************************************************************************** | ||
| # Copyright (C) 2022 Travis Scrimshaw <tcscrims at gmail.com> | ||
| # | ||
| # This program is free software: you can redistribute it and/or modify | ||
| # it under the terms of the GNU General Public License as published by | ||
| # the Free Software Foundation, either version 2 of the License, or | ||
| # (at your option) any later version. | ||
| # https://www.gnu.org/licenses/ | ||
| # **************************************************************************** | ||
|
|
||
| from sage.misc.cachefunc import cached_method | ||
| from sage.sets.family import Family | ||
| from sage.rings.infinity import infinity | ||
| from sage.rings.integer_ring import ZZ | ||
| from sage.categories.algebras import Algebras | ||
| from sage.combinat.free_module import CombinatorialFreeModule | ||
| from sage.monoids.free_abelian_monoid import FreeAbelianMonoid | ||
| from sage.matrix.constructor import matrix | ||
| from sage.structure.element import Matrix | ||
|
|
||
| class qCommutingPolynomials(CombinatorialFreeModule): | ||
| r""" | ||
| The algebra of `q`-commuting polynomials. | ||
| Let `R` be a commutative ring, and fix an element `q \in R`. Let | ||
| B = (B_{xy})_{x,y \in I}` be a skew-symmetric bilinear form with | ||
| index set `I`. Let `R[I]_{q,B}` denote the polynomial ring in the variables | ||
| `I` such that we have the `q`-*commuting* relation for `x, y \in I`: | ||
| .. MATH:: | ||
| y x = q^{B_{xy}} \cdot x y. | ||
| This is a graded `R`-algebra with a natural basis given by monomials | ||
| written in increasing order with respect to some total order on `I`. | ||
| When `B_{xy} = 1` and `B_{yx} = -1` for all `x < y`, then we have | ||
| a `q`-analog of the classical binomial coefficient theorem: | ||
| .. MATH:: | ||
| (x + y)^n = \sum_{k=0}^n \binom{n}{k}_q x^k y^{n-k}. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y> = algebras.qCommutingPolynomials(q) | ||
| We verify a case of the `q`-binomial theorem:: | ||
| sage: f = (x + y)^10 | ||
| sage: all(f[b] == q_binomial(10, b.list()[0]) for b in f.support()) | ||
| True | ||
| We now do a computation with a non-standard `B` matrix:: | ||
| sage: B = matrix([[0,1,2],[-1,0,3],[-2,-3,0]]) | ||
| sage: B | ||
| [ 0 1 2] | ||
| [-1 0 3] | ||
| [-2 -3 0] | ||
| sage: q = ZZ['q'].gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q, B) | ||
| sage: y * x | ||
| q*x*y | ||
| sage: z * x | ||
| q^2*x*z | ||
| sage: z * y | ||
| q^3*y*z | ||
| sage: f = (x + z)^10 | ||
| sage: all(f[b] == q_binomial(10, b.list()[0], q^2) for b in f.support()) | ||
| True | ||
| sage: f = (y + z)^10 | ||
| sage: all(f[b] == q_binomial(10, b.list()[1], q^3) for b in f.support()) | ||
| True | ||
| """ | ||
| @staticmethod | ||
| def __classcall_private__(cls, q, n=None, B=None, base_ring=None, names=None): | ||
| r""" | ||
| Normalize input to ensure a unique representation. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R1.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R2 = algebras.qCommutingPolynomials(q, base_ring=q.parent(), names='x,y,z') | ||
| sage: R3 = algebras.qCommutingPolynomials(q, names=['x', 'y', 'z']) | ||
| sage: R1 is R2 is R3 | ||
| True | ||
| """ | ||
| if base_ring is not None: | ||
| q = base_ring(q) | ||
|
|
||
| if B is None and isinstance(n, Matrix): | ||
| n, B = B, n | ||
|
|
||
| if names is None: | ||
| raise ValueError("the names of the variables must be given") | ||
| from sage.structure.category_object import normalize_names | ||
| if n is None: | ||
| if isinstance(names, str): | ||
| n = names.count(',') + 1 | ||
| else: | ||
| n = len(names) | ||
| names = normalize_names(n, names) | ||
| n = len(names) | ||
| if B is None: | ||
| B = matrix.zero(ZZ, n) | ||
| for i in range(n): | ||
| for j in range(i+1, n): | ||
| B[i,j] = 1 | ||
| B[j,i] = -1 | ||
| B.set_immutable() | ||
| else: | ||
| if not B.is_skew_symmetric(): | ||
| raise ValueError("the matrix must be skew symmetric") | ||
| B = B.change_ring(ZZ) | ||
| B.set_immutable() | ||
| return super().__classcall__(cls, q=q, B=B, names=names) | ||
|
|
||
| def __init__(self, q, B, names): | ||
| r""" | ||
| Initialize ``self``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: TestSuite(R).run() | ||
| """ | ||
| self._q = q | ||
| self._B = B | ||
| base_ring = q.parent() | ||
| indices = FreeAbelianMonoid(len(names), names) | ||
| category = Algebras(base_ring).WithBasis().Graded() | ||
| CombinatorialFreeModule.__init__(self, base_ring, indices, | ||
| bracket=False, prefix='', | ||
| sorting_key=qCommutingPolynomials._term_key, | ||
| names=indices.variable_names(), category=category) | ||
|
|
||
| def _repr_(self): | ||
| r""" | ||
| Return a string representation of ``self``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R | ||
| q-commuting polynomial ring in x, y, z over Fraction Field of | ||
| Univariate Polynomial Ring in q over Integer Ring with matrix: | ||
| [ 0 1 1] | ||
| [-1 0 1] | ||
| [-1 -1 0] | ||
| """ | ||
| names = ", ".join(self.variable_names()) | ||
| return "{}-commuting polynomial ring in {} over {} with matrix:\n{}".format(self._q, names, self.base_ring(), self._B) | ||
|
|
||
| def _latex_(self): | ||
| r""" | ||
| Return a latex representation of ``self``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: latex(R) | ||
| \mathrm{Frac}(\Bold{Z}[q])[x, y, z]_{q} | ||
| """ | ||
| from sage.misc.latex import latex | ||
| names = ", ".join(self.variable_names()) | ||
| return "{}[{}]_{{{}}}".format(latex(self.base_ring()), names, self._q) | ||
|
|
||
| @staticmethod | ||
| def _term_key(x): | ||
| r""" | ||
| Compute a key for ``x`` for comparisons. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: elt = (x*y^3*z^2).leading_support() | ||
| sage: R._term_key(elt) | ||
| (6, [2, 3, 1]) | ||
| """ | ||
| L = x.list() | ||
| L.reverse() | ||
| return (sum(L), L) | ||
|
|
||
| def gen(self, i): | ||
| r""" | ||
| Return the ``i``-generator of ``self``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R.gen(0) | ||
| x | ||
| sage: R.gen(2) | ||
| z | ||
| """ | ||
| return self.monomial(self._indices.gen(i)) | ||
|
|
||
| @cached_method | ||
| def gens(self): | ||
| r""" | ||
| Return the generators of ``self``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R.gens() | ||
| (x, y, z) | ||
| """ | ||
| return tuple([self.monomial(g) for g in self._indices.gens()]) | ||
|
|
||
| @cached_method | ||
| def algebra_generators(self): | ||
| r""" | ||
| Return the algebra generators of ``self``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R.algebra_generators() | ||
| Finite family {'x': x, 'y': y, 'z': z} | ||
| """ | ||
| d = {v: self.gen(i) for i,v in enumerate(self.variable_names())} | ||
| return Family(self.variable_names(), d.__getitem__, name="generator") | ||
|
|
||
| @cached_method | ||
| def one_basis(self): | ||
| r""" | ||
| Return the basis index of the element `1`. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R.one_basis() | ||
| 1 | ||
| """ | ||
| return self._indices.one() | ||
|
|
||
| def degree_on_basis(self, m): | ||
| r""" | ||
| Return the degree of the monomial index by ``m``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R.degree_on_basis(R.one_basis()) | ||
| 0 | ||
| sage: f = (x + y)^3 + z^3 | ||
| sage: f.degree() | ||
| 3 | ||
| """ | ||
| return sum(m.list()) | ||
|
|
||
| def dimension(self): | ||
| r""" | ||
| Return the dimension of ``self``, which is `\infty`. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q) | ||
| sage: R.dimension() | ||
| +Infinity | ||
| """ | ||
| return infinity | ||
|
|
||
| @cached_method | ||
| def product_on_basis(self, x, y): | ||
| r""" | ||
| Return the product of two monomials given by ``x`` and ``y``. | ||
| EXAMPLES:: | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y> = algebras.qCommutingPolynomials(q) | ||
| sage: R.product_on_basis(x.leading_support(), y.leading_support()) | ||
| x*y | ||
| sage: R.product_on_basis(y.leading_support(), x.leading_support()) | ||
| q*x*y | ||
| sage: x * y | ||
| x*y | ||
| sage: y * x | ||
| q*x*y | ||
| sage: y^2 * x | ||
| q^2*x*y^2 | ||
| sage: y * x^2 | ||
| q^2*x^2*y | ||
| sage: x * y * x | ||
| q*x^2*y | ||
| sage: y^2 * x^2 | ||
| q^4*x^2*y^2 | ||
| sage: (x + y)^2 | ||
| x^2 + (q+1)*x*y + y^2 | ||
| sage: (x + y)^3 | ||
| x^3 + (q^2+q+1)*x^2*y + (q^2+q+1)*x*y^2 + y^3 | ||
| sage: (x + y)^4 | ||
| x^4 + (q^3+q^2+q+1)*x^3*y + (q^4+q^3+2*q^2+q+1)*x^2*y^2 + (q^3+q^2+q+1)*x*y^3 + y^4 | ||
| With a non-standard `B` matrix:: | ||
| sage: B = matrix([[0,1,2],[-1,0,3],[-2,-3,0]]) | ||
| sage: q = ZZ['q'].fraction_field().gen() | ||
| sage: R.<x,y,z> = algebras.qCommutingPolynomials(q, B=B) | ||
| sage: x * y | ||
| x*y | ||
| sage: y * x^2 | ||
| q^2*x^2*y | ||
| sage: z^2 * x | ||
| q^4*x*z^2 | ||
| sage: z^2 * x^3 | ||
| q^12*x^3*z^2 | ||
| sage: z^2 * y | ||
| q^6*y*z^2 | ||
| sage: z^2 * y^3 | ||
| q^18*y^3*z^2 | ||
| """ | ||
| # Special case for multiplying by 1 | ||
| if x == self.one_basis(): | ||
| return self.monomial(y) | ||
| if y == self.one_basis(): | ||
| return self.monomial(x) | ||
|
|
||
| Lx = x.list() | ||
| Ly = y.list() | ||
|
|
||
| # This could be made more efficient | ||
| qpow = sum(exp * sum(self._B[j,i] * val for j, val in enumerate(Ly[:i])) for i,exp in enumerate(Lx)) | ||
| return self.term(x * y, self._q ** qpow) | ||
|
|