Orthogonal Symmetric Functions

AUTHORS:

• Travis Scrimshaw (2013-11-10): Initial version

class sage.combinat.sf.orthogonal.SymmetricFunctionAlgebra_orthogonal(Sym)

Bases: sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic

The orthogonal symmetric function basis (or orthogonal basis, to be short).

The orthogonal basis $\{o_{\lambda }\}$ where $\lambda$ is taken over all partitions is defined by the following change of basis with the Schur functions:

$$s_{\lambda }=\sum _{\mu }\left(\sum _{\nu \in H}{c}_{\mu \nu }^{\lambda }\right)o_{\mu }$$

where $H$ is the set of all partitions with even-width rows and $c_{\mu \nu }^{\lambda }$ 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 $\mathbf{Z}$-graded basis. However this does respect the induced $(\mathbf{Z}/2\mathbf{Z})$-grading.

INPUT:

• Sym – an instance of the ring of the symmetric functions

REFERENCES:

• [ChariKleber2000]

• [KoikeTerada1987]

• [ShimozonoZabrocki2006]

EXAMPLES:

Here are the first few orthogonal symmetric functions, in various bases:

sage: Sym = SymmetricFunctions(QQ)
sage: o = Sym.o()
sage: e = Sym.e()
sage: h = Sym.h()
sage: p = Sym.p()
sage: s = Sym.s()
sage: m = Sym.m()

sage: p(o([1]))
p[1]
sage: m(o([1]))
m[1]
sage: e(o([1]))
e[1]
sage: h(o([1]))
h[1]
sage: s(o([1]))
s[1]

sage: p(o([2]))
-p[] + 1/2*p[1, 1] + 1/2*p[2]
sage: m(o([2]))
-m[] + m[1, 1] + m[2]
sage: e(o([2]))
-e[] + e[1, 1] - e[2]
sage: h(o([2]))
-h[] + h[2]
sage: s(o([2]))
-s[] + s[2]

sage: p(o([3]))
-p[1] + 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3]
sage: m(o([3]))
-m[1] + m[1, 1, 1] + m[2, 1] + m[3]
sage: e(o([3]))
-e[1] + e[1, 1, 1] - 2*e[2, 1] + e[3]
sage: h(o([3]))
-h[1] + h[3]
sage: s(o([3]))
-s[1] + s[3]

sage: Sym = SymmetricFunctions(ZZ)
sage: o = Sym.o()
sage: e = Sym.e()
sage: h = Sym.h()
sage: s = Sym.s()
sage: m = Sym.m()
sage: p = Sym.p()
sage: m(o([4]))
-m[1, 1] + m[1, 1, 1, 1] - m[2] + m[2, 1, 1] + m[2, 2] + m[3, 1] + m[4]
sage: e(o([4]))
-e[1, 1] + e[1, 1, 1, 1] + e[2] - 3*e[2, 1, 1] + e[2, 2] + 2*e[3, 1] - e[4]
sage: h(o([4]))
-h[2] + h[4]
sage: s(o([4]))
-s[2] + s[4]

Some examples of conversions the other way:

sage: o(h[3])
o[1] + o[3]
sage: o(e[3])
o[1, 1, 1]
sage: o(m[2,1])
o[1] - 2*o[1, 1, 1] + o[2, 1]
sage: o(p[3])
o[1, 1, 1] - o[2, 1] + o[3]

Some multiplication:

sage: o([2]) * o([1,1])
o[1, 1] + o[2] + o[2, 1, 1] + o[3, 1]
sage: o([2,1,1]) * o([2])
o[1, 1] + o[1, 1, 1, 1] + 2*o[2, 1, 1] + o[2, 2] + o[2, 2, 1, 1]
 + o[3, 1] + o[3, 1, 1, 1] + o[3, 2, 1] + o[4, 1, 1]
sage: o([1,1]) * o([2,1])
o[1] + o[1, 1, 1] + 2*o[2, 1] + o[2, 1, 1, 1] + o[2, 2, 1]
 + o[3] + o[3, 1, 1] + o[3, 2]

Examples of the Hopf algebra structure:

sage: o([1]).antipode()
-o[1]
sage: o([2]).antipode()
-o[] + o[1, 1]
sage: o([1]).coproduct()
o[] # o[1] + o[1] # o[]
sage: o([2]).coproduct()
o[] # o[] + o[] # o[2] + o[1] # o[1] + o[2] # o[]
sage: o([1]).counit()
0
sage: o.one().counit()
1