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](1, 2) R. Orellana, M. Zabrocki, Symmetric group characters as symmetric functions, Arxiv 1510.00438.
class sage.combinat.sf.character.character_basis(Sym, other_basis, bname, pfix)

Bases: sage.combinat.sf.character.generic_character

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]
class sage.combinat.sf.character.generic_character(Sym, basis_name=None, prefix=None, graded=True)

Bases: sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic

class sage.combinat.sf.character.irreducible_character_basis(Sym, pfix)

Bases: sage.combinat.sf.character.generic_character

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