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== Q' Basis: == Timing data for arithmetic with HallLittlewood polynomials in the Q' basis. 

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Over ZZ:  == P basis == {{{ sage: HallLittlewoodP(QQ) HallLittlewood polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: HallLittlewoodP(QQ, t=1) HallLittlewood 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^2t)*s[1, 1, 1] + s[2, 1] }}} The HallLittlewood 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 HallLittlewood 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 HallLittlewood scalar product $<,>_t$. {{{ sage: HallLittlewoodQ(QQ) HallLittlewood polynomials in the Q basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: HallLittlewoodQ(QQ, t=1) HallLittlewood 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) HallLittlewood polynomials in the Qp basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: HallLittlewoodQp(QQ, t=1) HallLittlewood 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). 
HallLittlewood Polynomials
P basis
sage: HallLittlewoodP(QQ) HallLittlewood polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: HallLittlewoodP(QQ, t=1) HallLittlewood 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^2t)*s[1, 1, 1] + s[2, 1]
The HallLittlewood 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 HallLittlewood 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 HallLittlewood scalar product <,>_t.
sage: HallLittlewoodQ(QQ) HallLittlewood polynomials in the Q basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: HallLittlewoodQ(QQ, t=1) HallLittlewood 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) HallLittlewood polynomials in the Qp basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: HallLittlewoodQp(QQ, t=1) HallLittlewood 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