Orthogonal Symmetric Functions

AUTHORS:

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

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

Bases: 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:

\[s_{\lambda} = \sum_{\mu} \left( \sum_{\nu \in H} c^{\lambda}_{\mu\nu} \right) o_{\mu}\]

where \(H\) is the set of all partitions with even-width rows 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]

• [KoikeTerada1987]

• [ShimozonoZabrocki2006]

EXAMPLES:

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

Sage

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]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(QQ)
>>> o = Sym.o()
>>> e = Sym.e()
>>> h = Sym.h()
>>> p = Sym.p()
>>> s = Sym.s()
>>> m = Sym.m()

>>> p(o([Integer(1)]))
p[1]
>>> m(o([Integer(1)]))
m[1]
>>> e(o([Integer(1)]))
e[1]
>>> h(o([Integer(1)]))
h[1]
>>> s(o([Integer(1)]))
s[1]

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

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

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

Sage Live

Sym = SymmetricFunctions(QQ)
o = Sym.o()
e = Sym.e()
h = Sym.h()
p = Sym.p()
s = Sym.s()
m = Sym.m()
p(o([1]))
m(o([1]))
e(o([1]))
h(o([1]))
s(o([1]))
p(o([2]))
m(o([2]))
e(o([2]))
h(o([2]))
s(o([2]))
p(o([3]))
m(o([3]))
e(o([3]))
h(o([3]))
s(o([3]))
Sym = SymmetricFunctions(ZZ)
o = Sym.o()
e = Sym.e()
h = Sym.h()
s = Sym.s()
m = Sym.m()
p = Sym.p()
m(o([4]))
e(o([4]))
h(o([4]))
s(o([4]))

Some examples of conversions the other way:

Sage

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]

Python

>>> from sage.all import *
>>> o(h[Integer(3)])
o[1] + o[3]
>>> o(e[Integer(3)])
o[1, 1, 1]
>>> o(m[Integer(2),Integer(1)])
o[1] - 2*o[1, 1, 1] + o[2, 1]
>>> o(p[Integer(3)])
o[1, 1, 1] - o[2, 1] + o[3]

Sage Live

o(h[3])
o(e[3])
o(m[2,1])
o(p[3])

Some multiplication:

Sage

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]

Python

>>> from sage.all import *
>>> o([Integer(2)]) * o([Integer(1),Integer(1)])
o[1, 1] + o[2] + o[2, 1, 1] + o[3, 1]
>>> o([Integer(2),Integer(1),Integer(1)]) * o([Integer(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]
>>> o([Integer(1),Integer(1)]) * o([Integer(2),Integer(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]

Sage Live

o([2]) * o([1,1])
o([2,1,1]) * o([2])
o([1,1]) * o([2,1])

Examples of the Hopf algebra structure:

Sage

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

Python

>>> from sage.all import *
>>> o([Integer(1)]).antipode()
-o[1]
>>> o([Integer(2)]).antipode()
-o[] + o[1, 1]
>>> o([Integer(1)]).coproduct()
o[] # o[1] + o[1] # o[]
>>> o([Integer(2)]).coproduct()
o[] # o[] + o[] # o[2] + o[1] # o[1] + o[2] # o[]
>>> o([Integer(1)]).counit()
0
>>> o.one().counit()
1

Sage Live

o([1]).antipode()
o([2]).antipode()
o([1]).coproduct()
o([2]).coproduct()
o([1]).counit()
o.one().counit()

Make live