Symplectic Symmetric Functions#
AUTHORS:
Travis Scrimshaw (2013-11-10): Initial version
- class sage.combinat.sf.symplectic.SymmetricFunctionAlgebra_symplectic(Sym)#
Bases:
SymmetricFunctionAlgebra_generic
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:
\[s_{\lambda} = \sum_{\mu} \left( \sum_{\nu \in V} c^{\lambda}_{\mu\nu} \right) sp_{\mu}\]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