Hi all,
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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