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. |
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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 toother([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 (whereother
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]
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class
sage.combinat.sf.character.
generic_character
(Sym, basis_name=None, prefix=None, graded=True)¶ Bases:
sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic
Initializes the symmetric function algebra.
INPUT:
Sym
– the ring of symmetric functionsbasis_name
– name of basis (default:None
)prefix
– prefix used to display basisgraded
– (default:True
) ifTrue
, then the basis is considered to be graded, otherwise the basis is filtered
TESTS:
sage: from sage.combinat.sf.classical import SymmetricFunctionAlgebra_classical sage: s = SymmetricFunctions(QQ).s() sage: isinstance(s, SymmetricFunctionAlgebra_classical) True sage: TestSuite(s).run()
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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 tos([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
TESTS:
sage: TestSuite(st).run()