Differences between revisions 7 and 10 (spanning 3 versions)
 ⇤ ← Revision 7 as of 2008-03-28 01:05:51 → Size: 3367 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 1: Line 1: ## page was renamed from HallLittlewood Line 3: Line 2: This page contains preliminary results for doing computations with the Hall-Littlewood polynomials in the P, Q, and Q' bases. Line 5: Line 3: == P Basis: ==== Q Basis: ==== Q' Basis: ==Timing data for arithmetic with Hall-Littlewood polynomials in the Q' basis.=== Conversion to Schur basis ===Conversion to the Schur basis uses symmetrica.hall_littlewood. == P basis == Line 15: Line 5: sage: Qp = HallLittlewood_qp(QQ)sage: S = SFASchur(QQ['t'].fraction_field())sage: time b = S(Qp([2,2]))CPU times: user 0.05 s, sys: 0.00 s, total: 0.05 sWall time: 0.05sage: time b = S(Qp([3,2,1]))CPU times: user 0.16 s, sys: 0.00 s, total: 0.16 sWall time: 0.15sage: time b = S(Qp([3,3,2,1]))CPU times: user 0.87 s, sys: 0.03 s, total: 0.90 sWall time: 0.89sage: time b = S(Qp([2,2]))CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 sWall time: 0.02sage: time b = S(Qp([3,2,1]))CPU times: user 0.02 s, sys: 0.00 s, total: 0.03 sWall time: 0.03sage: time b = S(Qp([3,3,2,1]))CPU times: user 0.07 s, sys: 0.00 s, total: 0.07 sWall time: 0.07sage: bs[3, 3, 2, 1] + t*s[3, 3, 3] + t*s[4, 2, 2, 1] + (t^2+t)*s[4, 3, 1, 1] + (t^3+2*t^2+t)*s[4, 3, 2] + (t^4+t^3+t^2)*s[4, 4, 1] + (t^3+t^2)*s[5, 2, 1, 1] + (t^4+t^3+t^2)*s[5, 2, 2] + (t^5+2*t^4+3*t^3+t^2)*s[5, 3, 1] + (t^6+t^5+t^4+t^3)*s[5, 4] + t^4*s[6, 1, 1, 1] + (t^6+2*t^5+2*t^4+t^3)*s[6, 2, 1] + (t^7+t^6+2*t^5+2*t^4)*s[6, 3] + (t^7+t^6+t^5)*s[7, 1, 1] + (t^8+t^7+2*t^6+t^5)*s[7, 2] + (t^9+t^8+t^7)*s[8, 1] + t^10*s[9] 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] Line 39: Line 15: === Multiplication ===Multiplication in the Q' basis is performed by converting to the Schur basis, performing the multiplication there, and then converting back to the Q' basis. 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}}} Line 42: Line 23: Over ZZ: 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 45: 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 60: Line 51: 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 63: 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 77: Line 70: The majority of time spent in the last computation is due to coercion from ZZ['t'] to QQ('t') (which should really be much faster).Over RR:{{{sage: Qp = HallLittlewood_qp(RR)sage: time c = Qp([2,2])^2CPU times: user 0.78 s, sys: 0.01 s, total: 0.78 sWall time: 0.99sage: time c = Qp([3,2,1])^2CPU times: user 13.28 s, sys: 0.33 s, total: 13.61 sWall time: 13.67sage: time c = Qp([2,2])^2CPU times: user 0.44 s, sys: 0.00 s, total: 0.44 sWall time: 0.44sage: time c = Qp([3,2,1])^2CPU times: user 2.88 s, sys: 0.05 s, total: 2.94 sWall time: 2.95}}}The majority of time spent in the last computation is due to coercion from ZZ['t'] to RR('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)