Processing Math: Done
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Comment:
← Revision 11 as of 2008-11-14 13:42:03 → ⇥
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Editor: anonymous
Comment: converted to 1.6 markup
Deletions are marked like this. Additions are marked like this.
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== Q' Basis: ==
Timing data for arithmetic with Hall-Littlewood polynomials in the Q' basis.
Over ZZ:
== P basis ==
{{{
sage: HallLittlewoodP(QQ)
Hall-Littlewood polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: HallLittlewoodP(QQ, t=-1)
Hall-Littlewood polynomials in the P basis with t=-1 over Rational Field
sage: HLP = HallLittlewoodP(QQ)
sage: s = SFASchur(HLP.base_ring())
sage: s(HLP([2,1]))
(-t^2-t)*s[1, 1, 1] + s[2, 1]
}}}

The Hall-Littlewood polynomials in the P basis at t=0 are the Schur functions.
{{{
sage: HLP = HallLittlewoodP(QQ,t=0)
sage: s = SFASchur(HLP.base_ring())
sage: s(HLP([2,1])) == s([2,1])
True
}}}

The Hall-Littlewood polynomials in the P basis at t=1 are the monomial symmetric functions.
{{{
sage: HLP = HallLittlewoodP(QQ,t=1)
sage: m = SFAMonomial(HLP.base_ring())
sage: m(HLP([2,2,1])) == m([2,2,1])
True
}}}

== Q basis ==
The Q basis is dual to the P basis with respect to the Hall-Littlewood scalar product <,>t.
{{{
sage: HallLittlewoodQ(QQ)
Hall-Littlewood polynomials in the Q basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: HallLittlewoodQ(QQ, t=-1)
Hall-Littlewood polynomials in the Q basis with t=-1 over Rational Field
}}}
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sage: Qp = HallLittlewood_qp(ZZ)
sage: time c = Qp([2,2])^2
CPU times: user 0.54 s, sys: 0.01 s, total: 0.55 s
Wall time: 0.55
sage: time c = Qp([3,2,1])^2
CPU times: user 11.52 s, sys: 0.24 s, total: 11.76 s
Wall time: 11.78
sage: time c = Qp([2,2])^2
CPU times: user 0.21 s, sys: 0.01 s, total: 0.22 s
Wall time: 0.22
sage: time c = Qp([3,2,1])^2
CPU times: user 1.16 s, sys: 0.02 s, total: 1.18 s
Wall time: 1.18		 sage: HLP = HallLittlewoodP(QQ)
sage: HLQ = HallLittlewoodQ(QQ)
sage: HLP([2,1]).scalar_t(HLQ([2,1]))
1
sage: HLP([2,1]).scalar_t(HLQ([1,1,1]))
0
sage: HLP([2,1]).scalar_t(HLQ([3]))
0
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Over QQ:
== Qp basis ==
The Qp basis is dual to the P basis with respect to the standard Hall scalar product.
{{{
sage: HallLittlewoodQp(QQ)
Hall-Littlewood polynomials in the Qp basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: HallLittlewoodQp(QQ, t=-1)
Hall-Littlewood polynomials in the Qp basis with t=-1 over Rational Field
}}}
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sage: Qp = HallLittlewood_qp(QQ)
sage: time c = Qp([2,2])^2
CPU times: user 0.77 s, sys: 0.01 s, total: 0.78 s
Wall time: 0.78
sage: time c = Qp([3,2,1])^2
CPU times: user 14.00 s, sys: 0.24 s, total: 14.24 s
Wall time: 14.26
sage: time c = Qp([2,2])^2
CPU times: user 0.55 s, sys: 0.01 s, total: 0.56 s
Wall time: 0.56
sage: time c = Qp([3,2,1])^2
CPU times: user 3.57 s, sys: 0.08 s, total: 3.65 s
Wall time: 3.66		 sage: HLP = HallLittlewoodP(QQ)
sage: HLQp = HallLittlewoodQp(QQ)
sage: HLP([2,1]).scalar(HLQp([2,1]))
1
sage: HLP([2,1]).scalar(HLQp([1,1,1]))
0
sage: HLP([2,1]).scalar(HLQp([3]))
0
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The majority of time spent in the second one is due to coercion from ZZ['t'] to QQ('t') (which should really be much faster).

Hall-Littlewood Polynomials

P basis

sage: HallLittlewoodP(QQ)
Hall-Littlewood polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: HallLittlewoodP(QQ, t=-1)
Hall-Littlewood polynomials in the P basis with t=-1 over Rational Field
sage: HLP = HallLittlewoodP(QQ)
sage: s = SFASchur(HLP.base_ring())
sage: s(HLP([2,1]))
(-t^2-t)*s[1, 1, 1] + s[2, 1]

The Hall-Littlewood polynomials in the P basis at t=0 are the Schur functions. 

sage: HLP = HallLittlewoodP(QQ,t=0)
sage: s = SFASchur(HLP.base_ring())
sage: s(HLP([2,1])) == s([2,1])
True

The Hall-Littlewood polynomials in the P basis at t=1 are the monomial symmetric functions. 

sage: HLP = HallLittlewoodP(QQ,t=1)
sage: m = SFAMonomial(HLP.base_ring())
sage: m(HLP([2,2,1])) == m([2,2,1])
True

Q basis

The Q basis is dual to the P basis with respect to the Hall-Littlewood scalar product <,>t. 

sage: HallLittlewoodQ(QQ)
Hall-Littlewood polynomials in the Q basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: HallLittlewoodQ(QQ, t=-1)
Hall-Littlewood polynomials in the Q basis with t=-1 over Rational Field

sage: HLP = HallLittlewoodP(QQ)
sage: HLQ = HallLittlewoodQ(QQ)
sage: HLP([2,1]).scalar_t(HLQ([2,1]))
1
sage: HLP([2,1]).scalar_t(HLQ([1,1,1]))
0
sage: HLP([2,1]).scalar_t(HLQ([3]))
0

Qp basis

The Qp basis is dual to the P basis with respect to the standard Hall scalar product. 

sage: HallLittlewoodQp(QQ)
Hall-Littlewood polynomials in the Qp basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: HallLittlewoodQp(QQ, t=-1)
Hall-Littlewood polynomials in the Qp basis with t=-1 over Rational Field

sage: HLP = HallLittlewoodP(QQ)
sage: HLQp = HallLittlewoodQp(QQ)
sage: HLP([2,1]).scalar(HLQp([2,1]))
1
sage: HLP([2,1]).scalar(HLQp([1,1,1]))
0
sage: HLP([2,1]).scalar(HLQp([3]))
0

combinat/HallLittlewood (last edited 2008-11-14 13:42:03 by anonymous)