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== 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 }}} 
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