"""Section 23: on U_corr (inside the maximal unipotent of SO_{2m+1}, m=2n) the function
Lambda_c = sum g_{i,i+2} - a sum g_{2j,2j+1} + g_{m-1,-m} + (c/2) a   is additive (n=2, symbolic c),
and on U_alt the linear form sum g_{i,i+2} + g_{m-1,-m} + (c/2) g_{m-1,m} is additive."""
import sympy as sp
n=2; m=2*n; N=2*m+1
E_=lambda i:i-1; F_=lambda i:m+i-1; U=2*m
def Em(a,b):
    M=sp.zeros(N,N); M[a,b]=1; return M
def rv(i,j,kind):
    if kind=='-': return Em(E_(i),E_(j))-Em(F_(j),F_(i))
    if kind=='+': return Em(E_(i),F_(j))-Em(E_(j),F_(i))
    return 2*Em(E_(i),U)-Em(U,F_(i))
def expm(X):
    out=sp.eye(N); P=sp.eye(N); k=1
    while True:
        P=P*X/k; k+=1
        if P==sp.zeros(N,N): break
        out+=P
    return out
c=sp.Symbol('c')
def generic(tag,kind):
    a=sp.Symbol('a'+tag); g=sp.eye(N); cnt=0
    if kind=='corr':
        for j in range(1,n+1): g=g*expm(a*rv(2*j-1,2*j,'-'))
    for i in range(1,m+1):
        for j in range(i+1,m+1):
            tied=(j==i+1 and i%2==1)
            if kind=='corr' and tied: pass
            else:
                if not (kind=='alt' and j==i+1 and i%2==0):
                    cnt+=1; g=g*expm(sp.Symbol(f't{tag}{cnt}')*rv(i,j,'-'))
            cnt+=1; g=g*expm(sp.Symbol(f't{tag}{cnt}')*rv(i,j,'+'))
        if i!=m:
            cnt+=1; g=g*expm(sp.Symbol(f't{tag}{cnt}')*rv(i,None,'s'))
    return g.applyfunc(sp.expand)
def Lam(g):
    a=g[E_(1),E_(2)]
    return sum(g[E_(i),E_(i+2)] for i in range(1,m-1))-a*sum(g[E_(2*j),E_(2*j+1)] for j in range(1,n))+g[E_(m-1),F_(m)]+c/2*a
def Lalt(g):
    return sum(g[E_(i),E_(i+2)] for i in range(1,m-1))+g[E_(m-1),F_(m)]+c/2*g[E_(m-1),E_(m)]
g1=generic('x','corr'); g2=generic('y','corr'); p=(g1*g2).applyfunc(sp.expand)
tied=all(sp.expand(p[E_(2*j-1),E_(2*j)]-p[E_(1),E_(2)])==0 for j in range(1,n+1)) and p[E_(m),U]==0
print('U_corr closed under products:',tied,'  Lambda_c additive:',sp.expand(Lam(p)-Lam(g1)-Lam(g2))==0)
h1=generic('x','alt'); h2=generic('y','alt'); p=(h1*h2).applyfunc(sp.expand)
ok=all(p[E_(2*j),E_(2*j+1)]==0 for j in range(1,n)) and p[E_(m),U]==0
print('U_alt closed under products:',ok,'  its character additive:',sp.expand(Lalt(p)-Lalt(h1)-Lalt(h2))==0)