Differences between revisions 8 and 11 (spanning 3 versions)
 ⇤ ← Revision 8 as of 2008-03-28 01:08:58 → Size: 873 Editor: MikeHansen Comment: ← Revision 11 as of 2008-11-14 13:42:03 → ⇥ Size: 2028 Editor: anonymous Comment: converted to 1.6 markup Deletions are marked like this. Additions are marked like this. Line 30: Line 30: == 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 Fieldsage: 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]))1sage: HLP([2,1]).scalar_t(HLQ([1,1,1]))0sage: 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 Fieldsage: 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]))1sage: HLP([2,1]).scalar(HLQp([1,1,1]))0sage: HLP([2,1]).scalar(HLQp([3]))0}}}

# 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)