Hi Mark, Travis and Darij,
@ Travis: Thanks for catching the SR bug and for pointing me to the fact
pfaffian is now implemented in sage. Last time I checked (not so recently),
it wasn't:)
@ Mark: As I said, the denominators that *are not* power series are
vandermondes or C Weyl denominators. For vandermondes, I suppose I could do
the polynomial division in something (I don't know how to do it in sage and
obtain a polynomial per se, but I can certainly do it in mathematica. In
sage, it shouldn't be too difficult either) and then take the result, which
will be symmetric, and use that.
As a side note, uglier denominators might involve infinite q symbols like
\prod_{i<j} (t x_i x_j; q). This would be the Macdonald level. If infinite
q symbols are implemented in sage in a way compatible with symmetric
functions (I reason this might be hard), please let me know. Otherwise, I
suppose I can expand as a power series in q, which is a headache (given the
amount of infinite q symbols involved).
As for the ring, I want the very base ring/field to be R = Q(t) (or
Q(q,t)), as you suggested. But if you build S = R['x1,x2,y1,y2'], take a
poly f in here, lift it to symmetric functions (in the same 4 variables
over R) then how do you tell sage you want the coefficient of only the
schur function in the x's? (which again should be a nice factor times a
schur function in the y's).
@Darij: I didn't know about inject_variables(), but it looks like a neat
shorthand trick. Thanks.
db
On Tuesday, March 4, 2014 8:29:27 PM UTC+1, Dan Betea wrote:
>
>
>
> 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
>> >
>> > --
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>>
>>
>>
>>
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