On Tuesday, March 4, 2014 5:16:14 PM UTC+1, Mark Shimozono wrote:
>
> 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).
>
The denominators are mostly vandermondes. However, there is something in
type C (symplectic) which will have a vandermonde in the x's and a Weyl
type C denominator in the y's. But that's probably a subject for a
different email altogether, because multivariate Laurent power series are
not implemented in sage as far as I can tell.
>
> 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
> >
> > --
> > You received this message because you are subscribed to the Google
> Groups "sage-combinat-devel" group.
> > To unsubscribe from this group and stop receiving emails from it, send
> an email to sage-combinat-devel+unsubscr...@googlegroups.com <javascript:>.
>
> > To post to this group, send email to
> > sage-comb...@googlegroups.com<javascript:>.
>
> > Visit this group at http://groups.google.com/group/sage-combinat-devel.
> > For more options, visit https://groups.google.com/groups/opt_out.
>
>
>
>
>
>
--
You received this message because you are subscribed to the Google Groups
"sage-combinat-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-combinat-devel+unsubscr...@googlegroups.com.
To post to this group, send email to sage-combinat-devel@googlegroups.com.
Visit this group at http://groups.google.com/group/sage-combinat-devel.
For more options, visit https://groups.google.com/groups/opt_out.
