hodge_normalize.at

hodge version of normalizing a parameter

NOTE: the type hodgeParamPol is equal to [ParamPol]
however we cannot use any ParamPols here, because atlas
automatically finalizes parameters when put in a ParamPol
recall (see hodgeParamPol.at) if Q is a ParamPol,
dearrange(Q)=[(vec,Param)]:[(f1,p1),...,(fn,pn)] <-> f1(v)*p_1 + ... + fn(v)*p_n
we work entirely with these objects
the downside is there is no collection of terms
at the end, once all p_i are normal, do rearrange to get a hodgeParamPol (and collect terms)
n

I(x,lambda,nu=0) is a parameter
gamma=infinitesimal character
alpha: simple complex root
beta=theta_x(alpha)
nu=0 => <gamma,beta^v>=<gamma,alpha^v>
Suppose: <gamma,alpha^v><=0
  alpha simple, complex root (not necessarily integral)

a) If theta_x(alpha)<0 (alpha is a complex descent) then

I(x,lambda,0) -> I(s_alpha x,s_alpha\lambda,0)

b) If theta_x(alpha)>0 (\alpha is a complex ascent)
   AND <gamma,alpha^v>=0  then return I(x,lambda,0)

c) If theta_x(alpha)>0 (\alpha is a complex ascent)
   AND <gamma,alpha^v> < 0  (strict inequality)
let x'=s_alpha x,lambda'=s_alpha\lambda

I(x,lambda,0) -> vI(x',lambda,0) +
                 \sum_{k=1}^{k=[n/2]-1}v^{k-1}(v^2-1)I(x',lambda'-k\alpha,0) +
                 [v^{n/2}(v-1)I(x',lambda'-(n/2)\alpha,0)

The last term is the same as the general one except (v^2-1) is
replaced by (v-1), and it only occurs if n is even.