### Below is a Sage worksheet for our class on Symmetric Functions.
### Class webpage: https://www.math.ucla.edu/~galashin/206A_2022_sym_fcns.html
### To run the below code, create an account on cocalc.com, create a new Sage worksheet there, and copy the code.

#initialization:
Sym = SymmetricFunctions(QQ) #the ring of symmetric functions
m = Sym.m() #monomial
e = Sym.e() #elementary symmetric
h = Sym.h() #complete homogeneous
p = Sym.p() #power sum
s = Sym.s() #Schur functions
print("Init OK!")

print("To specify a basis element, supply a partition in the form [lambda_1,lambda_2,...] as an argument:")
h([2,2])
print("expanded in a different basis: ",h([2,2]),"=",m(h([2,2])))

print("Cycle through all partitions like this:")
n=3
for la in Partitions(n): #"lambda" is protected in python
	print(h(la),"=",m(h(la)))

print("You can also get directly the transition matrix:")
h.transition_matrix(m,n=3)

print("Expand a symmetric polynomial in some number of variables:")
poly=h([2]).expand(3) #specify the number of variables
print("h_2(x_0,x_1,x_2) =",poly)
print("Inverse operation is also sometimes useful:")
poly_inverse=Sym.from_polynomial(poly)
print(poly,"=",poly_inverse,"=",h(poly_inverse))

scalar1 = m([2,2]).scalar(h([2,2]))
scalar0 = m([2,2]).scalar(h([2,1,1]))

om = h([2,2]).omega()
#Below I'm using the percent notation for output formatting;
#see e.g. https://stackabuse.com/python-string-interpolation-with-the-percent-operator/
print("Scalar product: <%s,%s>=%s,   but <%s,%s>=%s"%(m([2,2]),h([2,2]),scalar1,m([2,2]),h([2,1,1]),scalar0))
print("Omega involution: omega(%s) = %s = %s"%(h([2,2]),om,e(om)))

print("Useful functions for partitions:")
la = Partition([5,3,1,1,1,0,0]) #this is not the same as [5,3,1,1,1,0,0], which is a list in python
print("lambda=",la)
print("size (sum of parts)=",la.size(),"; length (number of nonzero parts)=",la.length())
print("z_lambda = 1^3 * 3! * 3^1 * 1! * 5^1 * 1! = ",la.aut())
print("epsilon_lambda = (-1)^{number of even parts of lambda} =",la.sign())

print("Let's check that p_lambda's are orthogonal.")
n = 3
for la in Partitions(n):
	for mu in Partitions(n):
		print("<%s,%s>=%s"%(p(la),p(mu),p(la).scalar(p(mu))))
		if la == mu: print("z_lambda=",la.aut())

print("And now let's check that p_lambda's are eigenvectors of omega:")
n = 4
for la in Partitions(n):
	print("omega(%s) = %s; epsilon_%s = %s "%(p(la),p(la).omega(),la,la.sign()))
	if p(la).omega() != la.sign() * p(la): raise Exception("Theorem is false!")

print("Exercise: do the same for Schur functions and see what happens!")

# For more information, check these links:
# https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/partition.html
# http://sporadic.stanford.edu/reference/combinat/sage/combinat/sf/sfa.html

print("To get quick help on a function, place a question mark right next to it and evaluate:")
la=Partition([5,3,1,1,1])
Tableau(la.hook_lengths()).pp() #pp means "pretty print"
la.hook_lengths?