# 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

## 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