Coverage for local/lib/python2.7/site-packages/sage/combinat/sf/character.py : 100%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
""" Characters of the symmetric group as bases of the symmetric functions
Just as the Schur functions are the irreducible characters of `Gl_n` and form a basis of the symmetric functions, the irreducible symmetric group character basis are the irreducible characters of of `S_n` when the group is realized as the permutation matrices.
REFERENCES:
.. [OZ2015] \R. Orellana, M. Zabrocki, *Symmetric group characters as symmetric functions*, :arxiv:`1510.00438`. """
#***************************************************************************** # Copyright (C) 2015 Mike Zabrocki <zabrocki@mathstat.yorku.ca # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #*****************************************************************************
r""" A rank function for partitions.
The leading term of a homogeneous expression will be the partition with the largest key value.
This key value is `|\lambda|^2 + \lambda_0` and using the ``max`` function on a list of Partitions.
Of course it is possible that this rank function is equal for some partitions, but the leading term should be any one of these partitions.
INPUT:
- ``la`` -- a partition
OUTPUT:
- an integer
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: ht = SymmetricFunctions(QQ).ht() sage: ht._my_key(Partition([2,1,1])) 18 sage: ht._my_key(Partition([2,2])) 18 sage: ht._my_key(Partition([3,1])) 19 sage: ht._my_key(Partition([1,1,1,1])) 17 """ else:
r""" Convert an expression the target basis to the character-basis.
We use triangularity to determine the expansion by subtracting off the leading term. The target basis is specified by the method ``self._other``.
INPUT:
- ``sexpr`` -- an element of ``self._other`` basis
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: ht = SymmetricFunctions(QQ).ht() sage: h = Sym.h() sage: ht._other_to_self(h[2] + h([])) ht[] + ht[1] + ht[2] sage: st = SymmetricFunctions(QQ).st() sage: s = Sym.s() sage: st._other_to_self(s[1] + s([])) 2*st[] + st[1] sage: 7 * st[[]] * st[[]] 7*st[] """ remove_zeros=False)
r""" General code for a character basis (irreducible and induced trivial).
This is a basis of the symmetric functions that has the property that ``self(la).character_to_frobenius_image(n)`` is equal to ``other([n-sum(la)]+la)``.
It should also have the property that the (outer) structure constants are the analogue of the stable Kronecker coefficients on the ``other`` basis (where ``other`` is either the Schur or homogeneous bases).
These bases are introduced in [OZ2015]_.
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.s() sage: h = Sym.h() sage: ht = SymmetricFunctions(QQ).ht() sage: st = SymmetricFunctions(QQ).st() sage: ht(s[2,1]) ht[1, 1] + ht[2, 1] - ht[3] sage: s(ht[2,1]) s[1] - 2*s[1, 1] - 2*s[2] + s[2, 1] + s[3] sage: ht(h[2,1]) ht[1] + 2*ht[1, 1] + ht[2, 1] sage: h(ht[2,1]) h[1] - 2*h[1, 1] + h[2, 1] sage: st(ht[2,1]) st[] + 2*st[1] + st[1, 1] + 2*st[2] + st[2, 1] + st[3] sage: ht(st[2,1]) ht[1] - ht[1, 1] + ht[2, 1] - ht[3] sage: ht[2]*ht[1,1] ht[1, 1] + 2*ht[1, 1, 1] + ht[2, 1, 1] sage: h[4,2].kronecker_product(h[4,1,1]) h[2, 2, 1, 1] + 2*h[3, 1, 1, 1] + h[4, 1, 1] sage: s(st[2,1]) 3*s[1] - 2*s[1, 1] - 2*s[2] + s[2, 1] sage: st(s[2,1]) st[] + 3*st[1] + 2*st[1, 1] + 2*st[2] + st[2, 1] sage: st[2]*st[1] st[1] + st[1, 1] + st[2] + st[2, 1] + st[3] sage: s[4,2].kronecker_product(s[5,1]) s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2] + s[5, 1] """ r""" Initialize the basis and register coercions.
The coercions are set up between the ``other_basis``.
INPUT:
- ``Sym`` -- an instance of the symmetric function algebra - ``other_basis`` -- a basis of Sym - ``bname`` -- the name for this basis (convention: ends in "character") - ``pfix`` -- a prefix to use for the basis
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: ht = SymmetricFunctions(QQ).ht(); ht Symmetric Functions over Rational Field in the induced trivial character basis sage: st = SymmetricFunctions(QQ).st(); st Symmetric Functions over Rational Field in the irreducible symmetric group character basis sage: TestSuite(ht).run() """ codomain=self._other).register_as_coercion() self._other_to_self))
def _self_to_other_on_basis(self, lam): r""" Convert a character-basis element to the ``self._other`` basis.
This is a recursive procedure that is calculated by the assumption that the leading term of ``self(lam)`` is ``other(lam)`` and ``evalsf(self(lam),n) == other([n-sum(lam)]+lam)``.
INPUT:
- ``lam`` -- a partition
OUTPUT:
- an expression in the ``self._other`` basis
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: ht = SymmetricFunctions(QQ).ht() sage: ht._self_to_other_on_basis(Partition([2,1])) h[1] - 2*h[1, 1] + h[2, 1] sage: st = SymmetricFunctions(QQ).st() sage: st._self_to_other_on_basis(Partition([2,1])) 3*s[1] - 2*s[1, 1] - 2*s[2] + s[2, 1]
TESTS::
sage: h = SymmetricFunctions(QQ).h() sage: ht = SymmetricFunctions(QQ).ht() sage: st = SymmetricFunctions(QQ).st() sage: all(ht(h(ht[la])) == ht[la] for i in range(5) for la in Partitions(i)) True sage: all(h(ht(h[la])) == h[la] for i in range(5) for la in Partitions(i)) True sage: all(st(h(st[la])) == st[la] for i in range(5) for la in Partitions(i)) True sage: all(h(st(h[la])) == h[la] for i in range(5) for la in Partitions(i)) True """ for (mu,c) in sim if mu[1:] != lam)
r""" The irreducible symmetric group character basis of the symmetric functions.
This is a basis of the symmetric functions that has the property that ``self(la).character_to_frobenius_image(n)`` is equal to ``s([n-sum(la)]+la)``.
It should also have the property that the (outer) structure constants are the analogue of the stable kronecker coefficients on the Schur basis (where ``other`` is either the Schur or homogeneous bases).
This basis is introduced in [OZ2015]_.
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.s() sage: h = Sym.h() sage: ht = SymmetricFunctions(QQ).ht() sage: st = SymmetricFunctions(QQ).st() sage: st(ht[2,1]) st[] + 2*st[1] + st[1, 1] + 2*st[2] + st[2, 1] + st[3] sage: ht(st[2,1]) ht[1] - ht[1, 1] + ht[2, 1] - ht[3] sage: s(st[2,1]) 3*s[1] - 2*s[1, 1] - 2*s[2] + s[2, 1] sage: st(s[2,1]) st[] + 3*st[1] + 2*st[1, 1] + 2*st[2] + st[2, 1] sage: st[2]*st[1] st[1] + st[1, 1] + st[2] + st[2, 1] + st[3] sage: s[4,2].kronecker_product(s[5,1]) s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2] + s[5, 1] sage: st[1,1,1].counit() -1 sage: all(sum(c*st(la)*st(mu).antipode() for ....: ((la,mu),c) in st(ga).coproduct())==st(st(ga).counit()) ....: for ga in Partitions(3)) True
TESTS::
sage: TestSuite(st).run() """ r""" Initialize the basis and register coercions.
The coercions are set up between the ``other_basis``
INPUT:
- ``Sym`` -- an instance of the symmetric function algebra - ``pfix`` -- a prefix to use for the basis
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: ht = SymmetricFunctions(QQ).ht(); ht Symmetric Functions over Rational Field in the induced trivial character basis sage: st = SymmetricFunctions(QQ).st(); st Symmetric Functions over Rational Field in the irreducible symmetric group character basis """ basis_name="irreducible symmetric group character", prefix=pfix, graded=False)
codomain=Sym.powersum()).register_as_coercion() self._other_to_self))
r""" An expression involving moebius inversion in the powersum generators.
For a positive value of ``k``, this expression is
.. MATH::
\frac{1}{k} \sum_{d|k} \mu(d/k) p_d.
INPUT:
- ``k`` -- a positive integer
OUTPUT:
- an expression in the powersum basis of the symmetric functions
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st() sage: st._b_power_k(1) p[1] sage: st._b_power_k(2) -1/2*p[1] + 1/2*p[2] sage: st._b_power_k(6) 1/6*p[1] - 1/6*p[2] - 1/6*p[3] + 1/6*p[6]
""" for d in divisors(k))
r""" An expression involving moebius inversion in the powersum generators.
For a positive value of ``k``, this expression is
.. MATH::
\sum_{j=0}^r (-1)^{r-j}k^j\binom{r,j} \left( \frac{1}{k} \sum_{d|k} \mu(d/k) p_d \right)_k.
INPUT:
- ``k``, ``r`` -- positive integers
OUTPUT:
- an expression in the powersum basis of the symmetric functions
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st() sage: st._b_power_k_r(1,1) -p[] + p[1] sage: st._b_power_k_r(2,2) p[] + 4*p[1] + p[1, 1] - 4*p[2] - 2*p[2, 1] + p[2, 2] sage: st._b_power_k_r(3,2) p[] + 5*p[1] + p[1, 1] - 5*p[3] - 2*p[3, 1] + p[3, 3]
""" * p.prod(self._b_power_k(k) - i*p.one() for i in range(j)) for j in range(r+1) )
r""" An expression involving moebius inversion in the powersum generators.
For a partition `\gamma = (1^{m_1}, 2^{m_2}, \ldots r^{m_r})`, this expression is
.. MATH::
{\mathbf p}_{\ga} = \sum_{k \geq 1} {\mathbf p}_{k^{m_k}},
where
.. MATH::
{\mathbf p}_{k^r} = \sum_{j=0}^r (-1)^{r-j}k^j\binom{r,j} \left( \frac{1}{k} \sum_{d|k} \mu(d/k) p_d \right)_k~.
INPUT:
- ``gamma`` -- a partition
OUTPUT:
- an expression in the powersum basis of the symmetric functions
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st() sage: st._b_power_k_r(1,1) -p[] + p[1] sage: st._b_power_k_r(2,2) p[] + 4*p[1] + p[1, 1] - 4*p[2] - 2*p[2, 1] + p[2, 2] sage: st._b_power_k_r(3,2) p[] + 5*p[1] + p[1, 1] - 5*p[3] - 2*p[3, 1] + p[3, 3]
""" for (k,r) in six.iteritems(gamma.to_exp_dict()) )
r""" An expansion of the irreducible character in the powersum basis.
The formula for the irreducible character basis indexed by the partition ``lam`` is given by the formula
.. MATH::
\sum_{\gamma} \chi^{\lambda}(\gamma) \frac{{\mathbf p}_\gamma}{z_\gamma},
where `\chi^{\lambda}(\gamma)` is the irreducible character indexed by the partition `\lambda` and evaluated at an element of cycle structure `\gamma` and `{\mathbf p}_\gamma` is the power sum expression calculated in the method :meth:`_b_power_gamma`.
INPUT:
- ``lam`` -- a partition
OUTPUT:
- an expression in the power sum basis
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st() sage: st._self_to_power_on_basis([2,1]) 3*p[1] - 2*p[1, 1] + 1/3*p[1, 1, 1] - 1/3*p[3] sage: st._self_to_power_on_basis([1,1]) p[] - p[1] + 1/2*p[1, 1] - 1/2*p[2]
""" for (ga, c) in self._p(self._other(lam)) )
def _self_to_other_on_basis(self, lam): r""" An expansion of the irreducible character basis in the Schur basis.
Compute the Schur expansion by first computing it in the powersum basis and the coercing to the Schur basis.
INPUT:
- ``lam`` -- a partition
OUTPUT:
- an expression in the Schur basis
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st() sage: st._self_to_other_on_basis(Partition([1,1])) s[] - s[1] + s[1, 1] sage: st._self_to_other_on_basis(Partition([2,1])) 3*s[1] - 2*s[1, 1] - 2*s[2] + s[2, 1] """
|