Differences between revisions 1 and 10 (spanning 9 versions)
 ⇤ ← Revision 1 as of 2007-10-02 19:42:46 → Size: 1157 Editor: MikeHansen Comment: ← Revision 10 as of 2008-03-28 01:18:21 → ⇥ Size: 2028 Editor: MikeHansen Comment: Deletions are marked like this. Additions are marked like this. Line 2: Line 2: == Q' Basis: ==Timing data for arithmetic with Hall-Littlewood polynomials in the Q' basis.Over ZZ: == P basis =={{{sage: HallLittlewoodP(QQ)Hall-Littlewood polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Fieldsage: HallLittlewoodP(QQ, t=-1)Hall-Littlewood polynomials in the P basis with t=-1 over Rational Fieldsage: 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 Fieldsage: HallLittlewoodQ(QQ, t=-1)Hall-Littlewood polynomials in the Q basis with t=-1 over Rational Field}}} Line 7: Line 41: sage: Qp = HallLittlewood_qp(ZZ)sage: time c = Qp([2,2])^2CPU times: user 0.54 s, sys: 0.01 s, total: 0.55 sWall time: 0.55sage: time c = Qp([3,2,1])^2CPU times: user 11.52 s, sys: 0.24 s, total: 11.76 sWall time: 11.78sage: time c = Qp([2,2])^2CPU times: user 0.21 s, sys: 0.01 s, total: 0.22 sWall time: 0.22sage: time c = Qp([3,2,1])^2CPU times: user 1.16 s, sys: 0.02 s, total: 1.18 sWall time: 1.18 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 Line 21: Line 50: Over QQ: == 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}}} Line 24: Line 61: sage: Qp = HallLittlewood_qp(QQ)sage: time c = Qp([2,2])^2CPU times: user 0.77 s, sys: 0.01 s, total: 0.78 sWall time: 0.78sage: time c = Qp([3,2,1])^2CPU times: user 14.00 s, sys: 0.24 s, total: 14.24 sWall time: 14.26sage: time c = Qp([2,2])^2CPU times: user 0.55 s, sys: 0.01 s, total: 0.56 sWall time: 0.56sage: time c = Qp([3,2,1])^2CPU times: user 3.57 s, sys: 0.08 s, total: 3.65 sWall time: 3.66 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 Line 38: Line 70: 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).

# 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)