LLT Polynomials

sage.combinat.sf.llt.LLT(R, k, t=None)

Returns a class for working with LLT polynomials.

EXAMPLES:

sage: L3 = LLT(QQ,3); L3
LLT polynomials at level 3 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: L3.cospin([3,2,1])
(t+1)*m[1, 1] + m[2]
sage: L3.hcospin()
LLT polynomials in the HCosp basis at level 3 with t=t over Fraction Field of Univariate Polynomial Ring in t over Rational Field
class sage.combinat.sf.llt.LLTElement_cospin(A, x)
class sage.combinat.sf.llt.LLTElement_generic(A, x)
class sage.combinat.sf.llt.LLTElement_spin(A, x)
sage.combinat.sf.llt.LLTHCospin(R, level, t=None)

Returns the LLT polynomials in the HCospin basis at level level.

EXAMPLES:

sage: HCosp3 = LLTHCospin(QQ,3)
sage: HCosp3([1])^2
1/t*HCosp3[1, 1] + ((t-1)/t)*HCosp3[2]
sage.combinat.sf.llt.LLTHSpin(R, level, t=None)

Returns the LLT polynomials in the HSpin basis at level level.

EXAMPLES:

sage: HSp3 = LLTHSpin(QQ,3)
sage: HSp3([1])^2
HSp[1, 1] + (-t+1)*HSp[2]
class sage.combinat.sf.llt.LLT_class(R, k, t=None)
__cmp__(other)

EXAMPLES:

sage: L3Q = LLT(QQ,3)
sage: L4Q = LLT(QQ,4)
sage: L3Z = LLT(ZZ, 4)
sage: cmp(L3Q, L3Q)
0
sage: cmp(L3Q, L3Z)
-1
sage: cmp(L3Q, L4Q)
-1
sage: cmp(L3Q, QQ)
1
sage: cmp(LLT(QQ,3,2), LLT(QQ,3,3))
-1
__init__(R, k, t=None)

EXAMPLES:

sage: L3 = LLT(QQ,3)
sage: L3 == loads(dumps(L3))
True
__repr__()

EXAMPLES:

sage: LLT(QQ,3)
LLT polynomials at level 3 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: LLT(QQ,3,2)
LLT polynomials at level 3 with t=2 over Rational Field
_llt_generic(skp, stat)

Takes in partition, list of partitions, or a list of skew partitions as well as a statistic which takes in two partitions and a level and spits out a coefficient.

EXAMPLES:

sage: L3 = LLT(QQ,3)
sage: f = lambda skp,mu,level: QQ(1)
sage: L3._llt_generic([3,2,1],f)
m[1, 1] + m[2]
base_ring()

Returns the base ring of self.

EXAMPLES:

sage: LLT(QQ,3).base_ring()
Fraction Field of Univariate Polynomial Ring in t over Rational Field
cospin(skp)

EXAMPLES:

sage: L3 = LLT(QQ,3)
sage: L3.cospin([2,1])
m[1]
sage: L3.cospin([3,2,1])
(t+1)*m[1, 1] + m[2]
sage: s = SFASchur(L3.base_ring())
sage: s(L3.cospin([[2],[1],[2]]))
t^4*s[2, 2, 1] + t^3*s[3, 1, 1] + (t^3+t^2)*s[3, 2] + (t^2+t)*s[4, 1] + s[5]
hcospin()

Returns the HCopsin basis.

EXAMPLES:

sage: LLT(QQ,3).hcospin()
LLT polynomials in the HCosp basis at level 3 with t=t over Fraction Field of Univariate Polynomial Ring in t over Rational Field
hspin()

Returns the HSpin basis.

EXAMPLES:

sage: LLT(QQ,3).hspin()
LLT polynomials in the HSp basis at level 3 with t=t over Fraction Field of Univariate Polynomial Ring in t over Rational Field
level()

Returns the level of self.

EXAMPLES:

sage: LLT(QQ,3).level()
3
spin_square(skp)

Returns the spin polynomial associated with skp with the substitution t \rightarrow t^2 made.

EXAMPLES:

sage: L3 = LLT(QQ,3)
sage: L3.spin_square([2,1])
t*m[1]
sage: L3.spin_square([3,2,1])
(t^3+t)*m[1, 1] + t^3*m[2]
sage: L3.spin_square([[1],[1],[1]])
(t^6+2*t^4+2*t^2+1)*m[1, 1, 1] + (t^6+t^4+t^2)*m[2, 1] + t^6*m[3]
class sage.combinat.sf.llt.LLT_cospin(R, level, t=None)
__init__(R, level, t=None)

TESTS:

sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: HCosp3 == loads(dumps(HCosp3))
True
_coerce_start(x)

Coerce things into the LLT HCosp basis through the monomials.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: s = SFASchur(HCosp3.base_ring())
sage: HCosp3._coerce_start(s([2]))
HCosp3[2]
sage: HCosp3._coerce_start(s([1,1]))
1/t*HCosp3[1, 1] - 1/t*HCosp3[2]
_m_cache(n)

Compute the change of basis from the monomial symmetric functions to self.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: HCosp3._m_cache(2)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l( HCosp3._self_to_m_cache[2] )
[([1, 1], [([1, 1], t + 1), ([2], 1)]), ([2], [([1, 1], 1), ([2], 1)])]
sage: l( HCosp3._m_to_self_cache[2] )
[([1, 1], [([1, 1], 1/t), ([2], -1/t)]),
 ([2], [([1, 1], -1/t), ([2], (t + 1)/t)])]
_multiply(left, right)

Convert to the monomial basis, do the multiplication there, and convert back to the Hcosp basis.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: HCosp3([1])^2 #indirect doctest
1/t*HCosp3[1, 1] + ((t-1)/t)*HCosp3[2]
_to_m(part)

Returns a function which gives the coefficient of part2 in the monomial expansion of self(part).

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: f21 = HCosp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[1, t + 1, 2*t + 1]
class sage.combinat.sf.llt.LLT_generic(R, level, t=None)
__init__(R, level, t=None)

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: LLT_spin(QQ, 3)
LLT polynomials in the HSp basis at level 3 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: LLT_spin(QQ, 3, t=2)
LLT polynomials in the HSp basis at level 3 with t=2 over Rational Field
level()

Returns the level of self.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HSp3 = LLT_spin(QQ, 3)
sage: HSp3.level()
3
class sage.combinat.sf.llt.LLT_spin(R, level, t=None)
__init__(R, level, t=None)

TESTS:

sage: from sage.combinat.sf.llt import *
sage: HSp3 = LLT_spin(QQ, 3)
sage: HSp3 == loads(dumps(HSp3))
True
_coerce_start(x)

Coerce things into the LLT HSp basis through the monomials.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HSp3 = LLT_spin(QQ, 3)
sage: s = SFASchur(HSp3.base_ring())
sage: HSp3._coerce_start(s([2]))
HSp[2]
sage: HSp3._coerce_start(s([1,1]))
HSp[1, 1] - t*HSp[2]
_m_cache(n)

Compute the change of basis from the monomial symmetric functions to self.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HSp3 = LLT_spin(QQ, 3)
sage: HSp3._m_cache(2)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l( HSp3._self_to_m_cache[2] )
[([1, 1], [([1, 1], t + 1), ([2], t)]), ([2], [([1, 1], 1), ([2], 1)])]
sage: l( HSp3._m_to_self_cache[2] )
[([1, 1], [([1, 1], 1), ([2], -t)]), ([2], [([1, 1], -1), ([2], t + 1)])]
_multiply(left, right)

Convert to the monomial basis, do the multiplication there, and convert back to the HSp basis.

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HSp3 = LLT_spin(QQ, 3)
sage: HSp3([1])^2 #indirect doctest
HSp[1, 1] + (-t+1)*HSp[2]
_to_m(part)

Returns a function which gives the coefficient of a partition in the monomial expansion of self(part).

EXAMPLES:

sage: from sage.combinat.sf.llt import *
sage: HSp3 = LLT_spin(QQ, 3)
sage: f21 = HSp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[t, t + 1, t + 2]

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