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Deletions are marked like this.  Additions are marked like this. 
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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).  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). 
HallLittlewood Polynomials
Q' Basis:
Timing data for arithmetic with HallLittlewood polynomials in the Q' basis.
Conversion to Schur basis
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 s Wall time: 0.05 sage: time b = S(Qp([3,2,1])) CPU times: user 0.16 s, sys: 0.00 s, total: 0.16 s Wall time: 0.15 sage: time b = S(Qp([3,3,2,1])) CPU times: user 0.87 s, sys: 0.03 s, total: 0.90 s Wall time: 0.89 sage: time b = S(Qp([2,2])) CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s Wall time: 0.02 sage: time b = S(Qp([3,2,1])) CPU times: user 0.02 s, sys: 0.00 s, total: 0.03 s Wall time: 0.03 sage: time b = S(Qp([3,3,2,1])) CPU times: user 0.07 s, sys: 0.00 s, total: 0.07 s Wall time: 0.07 sage: b s[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]
Multiplication
Over ZZ:
sage: Qp = HallLittlewood_qp(ZZ) sage: time c = Qp([2,2])^2 CPU times: user 0.54 s, sys: 0.01 s, total: 0.55 s Wall time: 0.55 sage: time c = Qp([3,2,1])^2 CPU times: user 11.52 s, sys: 0.24 s, total: 11.76 s Wall time: 11.78 sage: time c = Qp([2,2])^2 CPU times: user 0.21 s, sys: 0.01 s, total: 0.22 s Wall time: 0.22 sage: time c = Qp([3,2,1])^2 CPU times: user 1.16 s, sys: 0.02 s, total: 1.18 s Wall time: 1.18
Over QQ:
sage: Qp = HallLittlewood_qp(QQ) sage: time c = Qp([2,2])^2 CPU times: user 0.77 s, sys: 0.01 s, total: 0.78 s Wall time: 0.78 sage: time c = Qp([3,2,1])^2 CPU times: user 14.00 s, sys: 0.24 s, total: 14.24 s Wall time: 14.26 sage: time c = Qp([2,2])^2 CPU times: user 0.55 s, sys: 0.01 s, total: 0.56 s Wall time: 0.56 sage: time c = Qp([3,2,1])^2 CPU times: user 3.57 s, sys: 0.08 s, total: 3.65 s Wall time: 3.66
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).