# Skew Schur function

I have worked with the Schur function using the following command for example

sage: Sym = SymmetricFunctions(QQ)
sage: P = Sym.p()  # to expand symmetric functions in power symmetric basis
sage: s = Sym.schur()
sage: P(s([2,2]))
1/12*p[1, 1, 1, 1] + 1/4*p[2, 2] + (-1/3)*p[3, 1]


Recently I need to work with the skew Schur function. I want to expand the shifted schur function $$s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y)$$ I am guessing that it can be the following identity $$s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y) = \sum_{\mu} s_{\lambda \setminus mu}(y) s_{\mu}(x_1 , x_2, \ldots, x_n)$$ For $y=1$ my idenity can be proven as I am sure there is an expression. The expression $s_{\lambda \setminus mu}(y)$ is skew Schur function. I am wondering if there are packages in sagemath where I can work with skew schur function and verify my identity is correct or not.

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I tried the following to check the identity for some specific $\lambda$ value, say $$\lambda = [4,3,2]\ .$$ Then the L.H.S. in the expression above, $$s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y)$$ is computed by the following code:

R.<y> = PolynomialRing(QQ)
Sym = SymmetricFunctions(R)
Sym.inject_shorthands()

lam = Partition([3, 2, 1])
slam6 = s[lam].expand(6)
slam6.parent().inject_variables()     # defining x0, x1, ... , x5

pol = slam6(*(xj + y for xj in (x0, x1, x2, x3, x4, x5)))
# pol    # decomment to see it


Of course, we want to have it as a Schur function, taken from the above polynomial, yes, it is the following (manually split) expression:

sage: s.from_polynomial(pol)
896*y^6*s[] + 896*y^5*s[1] + 448*y^4*s[1, 1] + 112*y^3*s[1, 1, 1] \
+ 320*y^4*s[2] + 192*y^3*s[2, 1] + 48*y^2*s[2, 1, 1] + 32*y^2*s[2, 2] \
+ 8*y*s[2, 2, 1] + 40*y^3*s[3] + 24*y^2*s[3, 1] + 6*y*s[3, 1, 1] \
+ 4*y*s[3, 2] + s[3, 2, 1]


Let us check the coefficient in $\mu=[3,1]$ from the R.H.S. of the claimed identity. We expect $24\;y^2\;s[3, 1]$ from the above expansion.

mu = Partition([3, 1])
skew = lam / mu
print(f'lam = {lam} , mu = {mu} , s(lam / mu) is {s(skew)}')


This gives

lam = [3, 2, 1] , mu = [3, 1] , s(lam / mu) is s[1, 1] + s[2]


We have to plug in now $y$ in the above Schur polynomial. If my plug in uses the right conventions, then we obtain a different result.

sage: s(skew).expand(6)(y,y,y,y,y,y)
36*y^2
sage: s(skew).expand(1)(y)
y^2


Here are the Schur polynomials for the skew partitions $\lambda/\mu$ for further partitions $\mu$:

for k in [0..6]:
for mu in Partitions(k):
if lam.contains(mu):
print('lam = %s :: mu = %-9s :: s(lam / mu) = %s' % (lam, mu, s(lam/mu)))


This gives:

lam = [3, 2, 1] :: mu = []        :: s(lam / mu) = s[3, 2, 1]
lam = [3, 2, 1] :: mu = [1]       :: s(lam / mu) = s[2, 2, 1] + s[3, 1, 1] + s[3, 2]
lam = [3, 2, 1] :: mu = [2]       :: s(lam / mu) = s[2, 1, 1] + s[2, 2] + s[3, 1]
lam = [3, 2, 1] :: mu = [1, 1]    :: s(lam / mu) = s[2, 1, 1] + s[2, 2] + s[3, 1]
lam = [3, 2, 1] :: mu = [3]       :: s(lam / mu) = s[2, 1]
lam = [3, 2, 1] :: mu = [2, 1]    :: s(lam / mu) = s[1, 1, 1] + 2*s[2, 1] + s[3]
lam = [3, 2, 1] :: mu = [1, 1, 1] :: s(lam / mu) = s[2, 1]
lam = [3, 2, 1] :: mu = [3, 1]    :: s(lam / mu) = s[1, 1] + s[2]
lam = [3, 2, 1] :: mu = [2, 2]    :: s(lam / mu) = s[1, 1] + s[2]
lam = [3, 2, 1] :: mu = [2, 1, 1] :: s(lam / mu) = s[1, 1] + s[2]
lam = [3, 2, 1] :: mu = [3, 2]    :: s(lam / mu) = s[1]
lam = [3, 2, 1] :: mu = [3, 1, 1] :: s(lam / mu) = s[1]
lam = [3, 2, 1] :: mu = [2, 2, 1] :: s(lam / mu) = s[1]
lam = [3, 2, 1] :: mu = [3, 2, 1] :: s(lam / mu) = s[]

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