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This page contains preliminary results for doing computations with the Hall-Littlewood polynomials in the P, Q, and Q' bases.
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== P Basis: ==

== Q Basis: ==

== Q' Basis: ==
Timing data for arithmetic with Hall-Littlewood polynomials in the Q' basis.

=== Conversion to Schur basis ===
Conversion to the Schur basis uses symmetrica.hall_littlewood.
== P basis ==
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sage: Qp = HallLittlewood_qp(QQ)
sage: S = SFASchur(QQ['t'].fraction_field())
sage: time b = S(Qp([2,2]))
CPU times: user 0.05 s, sys: 0.00 s, total: 0.05 s
Wall time: 0.05
sage: time b = S(Qp([3,2,1]))
CPU times: user 0.16 s, sys: 0.00 s, total: 0.16 s
Wall time: 0.15
sage: time b = S(Qp([3,3,2,1]))
CPU times: user 0.87 s, sys: 0.03 s, total: 0.90 s
Wall time: 0.89
sage: time b = S(Qp([2,2]))
CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s
Wall time: 0.02
sage: time b = S(Qp([3,2,1]))
CPU times: user 0.02 s, sys: 0.00 s, total: 0.03 s
Wall time: 0.03
sage: time b = S(Qp([3,3,2,1]))
CPU times: user 0.07 s, sys: 0.00 s, total: 0.07 s
Wall time: 0.07
sage: b
s[3, 3, 2, 1] + t*s[3, 3, 3] + t*s[4, 2, 2, 1] + (t^2+t)*s[4, 3, 1, 1] + (t^3+2*t^2+t)*s[4, 3, 2] + (t^4+t^3+t^2)*s[4, 4, 1] + (t^3+t^2)*s[5, 2, 1, 1] + (t^4+t^3+t^2)*s[5, 2, 2] + (t^5+2*t^4+3*t^3+t^2)*s[5, 3, 1] + (t^6+t^5+t^4+t^3)*s[5, 4] + t^4*s[6, 1, 1, 1] + (t^6+2*t^5+2*t^4+t^3)*s[6, 2, 1] + (t^7+t^6+2*t^5+2*t^4)*s[6, 3] + (t^7+t^6+t^5)*s[7, 1, 1] + (t^8+t^7+2*t^6+t^5)*s[7, 2] + (t^9+t^8+t^7)*s[8, 1] + t^10*s[9]
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]
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=== Multiplication ===
Multiplication in the Q' basis is performed by converting to the Schur basis, performing the multiplication there, and then converting back to the Q' basis.

Over ZZ:
The Hall-Littlewood polynomials in the P basis at $t = 0$ are the Schur functions.
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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,t=0)
sage: s = SFASchur(HLP.base_ring())
sage: s(HLP([2,1])) == s([2,1])
True
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Over QQ:
The Hall-Littlewood polynomials in the P basis at $t = 1$ are the monomial symmetric functions.
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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,t=1)
sage: m = SFAMonomial(HLP.base_ring())
sage: m(HLP([2,2,1])) == m([2,2,1])
True
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The majority of time spent in the last computation is due to coercion from ZZ['t'] to QQ('t') (which should really be much faster).


Over RR:

{{{
sage: Qp = HallLittlewood_qp(RR)
sage: time c = Qp([2,2])^2
CPU times: user 0.78 s, sys: 0.01 s, total: 0.78 s
Wall time: 0.99
sage: time c = Qp([3,2,1])^2
CPU times: user 13.28 s, sys: 0.33 s, total: 13.61 s
Wall time: 13.67
sage: time c = Qp([2,2])^2
CPU times: user 0.44 s, sys: 0.00 s, total: 0.44 s
Wall time: 0.44
sage: time c = Qp([3,2,1])^2
CPU times: user 2.88 s, sys: 0.05 s, total: 2.94 s
Wall time: 2.95
}}}
The majority of time spent in the last computation is due to coercion from ZZ['t'] to RR('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

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