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""" Symplectic Symmetric Functions
AUTHORS:
- Travis Scrimshaw (2013-11-10): Initial version """ #***************************************************************************** # Copyright (C) 2013 Travis Scrimshaw <tscrim at ucdavis.edu> # # 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""" The symplectic symmetric function basis (or symplectic basis, to be short).
The symplectic basis `\{ sp_{\lambda} \}` where `\lambda` is taken over all partitions is defined by the following change of basis with the Schur functions:
.. MATH::
s_{\lambda} = \sum_{\mu} \left( \sum_{\nu \in V} c^{\lambda}_{\mu\nu} \right) sp_{\mu}
where `V` is the set of all partitions with even-height columns and `c^{\lambda}_{\mu\nu}` is the usual Littlewood-Richardson (LR) coefficients. By the properties of LR coefficients, this can be shown to be a upper unitriangular change of basis.
.. NOTE::
This is only a filtered basis, not a `\ZZ`-graded basis. However this does respect the induced `(\ZZ/2\ZZ)`-grading.
INPUT:
- ``Sym`` -- an instance of the ring of the symmetric functions
REFERENCES:
.. [ChariKleber2000] Vyjayanthi Chari and Michael Kleber. *Symmetric functions and representations of quantum affine algebras*. :arXiv:`math/0011161v1`
.. [KoikeTerada1987] \K. Koike, I. Terada, *Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn*. J. Algebra 107 (1987), no. 2, 466-511.
.. [ShimozonoZabrocki2006] Mark Shimozono and Mike Zabrocki. *Deformed universal characters for classical and affine algebras*. Journal of Algebra, **299** (2006). :arxiv:`math/0404288`.
EXAMPLES:
Here are the first few symplectic symmetric functions, in various bases::
sage: Sym = SymmetricFunctions(QQ) sage: sp = Sym.sp() sage: e = Sym.e() sage: h = Sym.h() sage: p = Sym.p() sage: s = Sym.s() sage: m = Sym.m()
sage: p(sp([1])) p[1] sage: m(sp([1])) m[1] sage: e(sp([1])) e[1] sage: h(sp([1])) h[1] sage: s(sp([1])) s[1]
sage: p(sp([2])) 1/2*p[1, 1] + 1/2*p[2] sage: m(sp([2])) m[1, 1] + m[2] sage: e(sp([2])) e[1, 1] - e[2] sage: h(sp([2])) h[2] sage: s(sp([2])) s[2]
sage: p(sp([3])) 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3] sage: m(sp([3])) m[1, 1, 1] + m[2, 1] + m[3] sage: e(sp([3])) e[1, 1, 1] - 2*e[2, 1] + e[3] sage: h(sp([3])) h[3] sage: s(sp([3])) s[3]
sage: Sym = SymmetricFunctions(ZZ) sage: sp = Sym.sp() sage: e = Sym.e() sage: h = Sym.h() sage: s = Sym.s() sage: m = Sym.m() sage: p = Sym.p() sage: m(sp([4])) m[1, 1, 1, 1] + m[2, 1, 1] + m[2, 2] + m[3, 1] + m[4] sage: e(sp([4])) e[1, 1, 1, 1] - 3*e[2, 1, 1] + e[2, 2] + 2*e[3, 1] - e[4] sage: h(sp([4])) h[4] sage: s(sp([4])) s[4]
Some examples of conversions the other way::
sage: sp(h[3]) sp[3] sage: sp(e[3]) sp[1] + sp[1, 1, 1] sage: sp(m[2,1]) -sp[1] - 2*sp[1, 1, 1] + sp[2, 1] sage: sp(p[3]) sp[1, 1, 1] - sp[2, 1] + sp[3]
Some multiplication::
sage: sp([2]) * sp([1,1]) sp[1, 1] + sp[2] + sp[2, 1, 1] + sp[3, 1] sage: sp([2,1,1]) * sp([2]) sp[1, 1] + sp[1, 1, 1, 1] + 2*sp[2, 1, 1] + sp[2, 2] + sp[2, 2, 1, 1] + sp[3, 1] + sp[3, 1, 1, 1] + sp[3, 2, 1] + sp[4, 1, 1] sage: sp([1,1]) * sp([2,1]) sp[1] + sp[1, 1, 1] + 2*sp[2, 1] + sp[2, 1, 1, 1] + sp[2, 2, 1] + sp[3] + sp[3, 1, 1] + sp[3, 2]
Examples of the Hopf algebra structure::
sage: sp([1]).antipode() -sp[1] sage: sp([2]).antipode() sp[] + sp[1, 1] sage: sp([1]).coproduct() sp[] # sp[1] + sp[1] # sp[] sage: sp([2]).coproduct() sp[] # sp[2] + sp[1] # sp[1] + sp[2] # sp[] sage: sp([1]).counit() 0 sage: sp.one().counit() 1 """ """ Initialize ``self``.
TESTS::
sage: sp = SymmetricFunctions(QQ).sp() sage: TestSuite(sp).run() """ 'sp', graded=False)
# We make a strong reference since we use it for our computations # and so we can define the coercion below (only codomains have # strong references)
# Setup the coercions triangular='upper', unitriangular=True) triangular='upper', unitriangular=True)
def _sp_to_s_on_basis(self, lam): r""" Return the symplectic symmetric function ``sp[lam]`` expanded in the Schur basis, where ``lam`` is a partition.
TESTS:
Check that this is the inverse::
sage: sp = SymmetricFunctions(QQ).sp() sage: s = SymmetricFunctions(QQ).s() sage: all(sp(s(sp[la])) == sp[la] for i in range(5) for la in Partitions(i)) True sage: all(s(sp(s[la])) == s[la] for i in range(5) for la in Partitions(i)) True """ for nu in Partitions(2*j) if all(nu.leg_length(i,i) == nu.arm_length(i,i)+1 for i in range(nu.frobenius_rank())) ) for j in range(n//2+1) # // 2 for horizontal dominoes for mu in Partitions(n-2*j) })
def _s_to_sp_on_basis(self, lam): r""" Return the Schur symmetric function ``s[lam]`` expanded in the symplectic basis, where ``lam`` is a partition.
INPUT:
- ``lam`` -- a partition
OUTPUT:
- the expansion of ``s[lam]`` in the symplectic basis ``self``
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: s = Sym.schur() sage: sp = Sym.symplectic() sage: sp._s_to_sp_on_basis(Partition([])) sp[] sage: sp._s_to_sp_on_basis(Partition([4,2,1])) sp[2, 1] + sp[3] + sp[3, 1, 1] + sp[3, 2] + sp[4, 1] + sp[4, 2, 1] sage: s(sp._s_to_sp_on_basis(Partition([3,1]))) == s[3,1] True """ for nu in Partitions(j) ) for j in range(n//2+1) # // 2 for vertical dominoes for mu in Partitions(n-2*j) })
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