Atlas of Lie Groups and Representations

Documentation contents

atlas Library · Script File

is_normal.at

Two different definitions of normal:

1)Parameter p=(x,lambda,nu) is normal if
gamma is (weakly) dominant, and all singular complex roots are type C+,
i.e. dominant and no complex descents

2)a KType mu=(x,lambda), i.e. parameter with nu=0, i.e. a
standardrepk, is normal if (see standardrepk.h):

Normal: $\<\alpha^\vee(1+\theta),\lambda>\geq0$ when $\alpha$ simple,
complex, and orthogonal to sums of positive imaginary resp. real
roots.
Source
atlas-scripts/is_normal.at (74 lines)
Definitions
7
Loads
Loaded by
hodge_normalize.at K_type_formula.at

Definitions in source order

RC_simple

L23RC_simple(KGBElt x) = [int]
this file defines
is_normal@Param
is_normal_K@KType
the different names aren't necessary but are to avoid accidentally using the wrong one

is_normal_K

L35is_normal_K(KType mu) = (bool,int)
return (true,-1) or (false,witness)

also in this file at line 46

monomials

L42monomials([(vec,KType)] Q) = [KType]
note: can be multiplicities here

also defined in basic.at, hodgeParamLaurentPol.at

is_normal_K

L46is_normal_K([(vec,KType)] Q) = (bool,int)
return (true,-1) if all terms are normal, otherwise
return is_normal(p)=(false,witness) for some p in monomials[Q]

also in this file at line 35

is_normal 3 overloads

L54is_normal(Param p) = (bool,int)
normal: integrally dominant and no singular complex descents
correction: normal: dominant and no singular complex descents
L65is_normal ([(vec,Param)] Q) = (bool,int,int)
return (bool,# of term, #of simple coroot)
L72is_normal ([int] Levi, Param p)
For parameters on a general (not necessarily theta-stable) Levi

Generated from atlas-scripts at commit 648506c (2026-10-13).