Dan,
Before giving any "sage advice" I need to know how bad the
"real denominators" will get.
A special trick can be used if the denominators are limited to
(x_i - x_j).
Your toy example involves (1 - t x_i y_j)^{-1}
which is no problem at all using geometric series in an extra variable.
--Mark
> Sorry for asking a somewhat related question to one I asked a while ago,
> but I find myself in the situation of trying to expand a certain
> multivariate symmetric series (polynomial after cutoff) in certain classes
> of symmetric functions and can't seem to get it working.
>
> The following is true:
>
> \sum_{\lambda} \product_{i=1...n} (1-t^{\lambda_i + n - i + 1}) s_{\lambda}
> (x_1,...,x_n) s_{\lambda} (y_1,...,y_n) = \\
>
> \frac {1} {\prod_{i < j} (x_i - x_j) (y_i - y_j)} \det_{i,j} ( \frac {1-t}
> {(1-t x_i y_j) (1-x_i y_j)} )
>
> Proof is simple, but generalizations to Hall Littlewood or Macdonald
> polynomials (with non trivial proofs) exist due to Warnaar and
> Noumi-Kirillov.
>
> Question: Suppose I don't know the identity but suspect something along
> those lines is true. How do I expand the right hand side (a quotient of a
> determinant by the 2 vandermondes) in Schur functions (in x's say) and
> recover the coefficients (schur functions in y's times some products of 1 -
> q^? )
>
> I tried the following naive idea (suggested some time ago in a slightly
> different context) and failed. Basically, using power series rings.
>
> n=2 # number of variables
> # setup of the rings
> R1.<t> = QQ[]
> R1 = FractionField(R1)
> dp = n+4
> R = PowerSeriesRing(R1, n, 'z', default_prec = dp)
> S = PowerSeriesRing(R, n, 'x', default_prec=dp)
> Sym = SymmetricFunctions(R)
>
> # set up the determinant divided by the vandermondes
> M = matrix(SR,2, [(1-R1.gen(0)) / ( (1 -
> R1.gen(0)*S.gen(0)*S.base_ring().gen(0)) * (1 -
> S.gen(0)*S.base_ring().gen(0)) ), (1-R1.gen(0)) / ( (1 -
> R1.gen(0)*S.gen(0)*S.base_ring().gen(1)) * (1 -
> S.gen(0)*S.base_ring().gen(1)) ), (1-R1.gen(0)) / ( (1 -
> R1.gen(0)*S.gen(1)*S.base_ring().gen(0)) * (1 -
> S.gen(1)*S.base_ring().gen(0)) ), (1-R1.gen(0)) / ( (1 -
> R1.gen(0)*S.gen(1)*S.base_ring().gen(1)) * (1 -
> S.gen(1)*S.base_ring().gen(1)) )])
>
> h = M.det()
> h *= prod( 1/(S.gen(i) - S.gen(j)) for i in range(n) for j in range(i+1,n))
> h *= prod( 1/(S.base_ring().gen(i) - S.base_ring().gen(j)) for i in
> range(n) for j in range(i+1,n))
>
> Schur = Sym.schur()
>
> #do the expansion
> Schur_det_expansion = Schur(Sym.from_polynomial(h.polynomial()))
>
> # for a small partition lambda print the coefficient
> la = Partition([2,1])
> print Schur_det_expansion.coefficient(la)
>
> The errors I get are varied. If I do the determinant first (the case
> above), there's an error saying symbolic ring is not the same as the power
> series ring (I suppose the determinant returns an element of symbolic ring
> and sees a power series inside). If I divide by the vandermondes first,
> well those are not power series so the error claims I can only divide by a
> unit (and x_i - x_j ain't a unit). But the two errors of course cancel each
> other in theory (the vandermondes divide the determinant) and the end
> result makes sense as exemplified by the theorem I stated.
>
> I was deliberately stating this problem in the simplest case to avoid
> cumbersome notations, but the next step is to go to Hall-Littlewood and
> beyond.
>
> As a side question, are there routines that compute Pfaffians in sage?
>
> Can anybody help? Please?:)
>
> Thanks,
>
> Dan
>
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