Atlas of Lie Groups and Representations

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Script File

dirac_index.at

Dirac index for equal rank groups

also G(R), i.e. K, should be connected. Otherwise
you will get an answer, somehow related to the identity
component, but it's precise meaning isn't clear
Source
atlas-scripts/dirac_index.at (92 lines)
Definitions
13
Loads
Loaded by
none of the other all.at files

Definitions in source order

dv

L8dv = true
dv standard for dirac_verbose: turn debugging on/off
two-fold cover of G(C) defined by \gamma\in (1/2)X^*
L10
X*(rd)=Z^n -> lattice <Z^n,v>\subset (1/2)Z^n
where 2v is integral
the new lattice is the span of id_mat(rank)#v
 compute this by multiplying by 2, using image_lattice, and dividing by 2
 this gives a rational matrix A_i, whose inverse A is integral
simple_roots in this new basis: A*rd.simple_roots
simple_coroots in the new dual basis: ^A_i*rd.simple_coroots
new root datum has (roots,coroots)=(A*rd.simple_roots,^A_i*rd.simple_coroots)
note that (^roots)*coroots=^(A*rd.simple_roots)*(^A_i*rd.simple_coroots)
 =rd.simple_roots*(^A*^A_i)*rd.simple_coroots = original Cartan matrix
L11

cover

L23cover(RootDatum rd,ratvec v) = (RootDatum,mat)
mat:A takes roots for rd->roots for cover
L31

cover_c

L34cover_c(RootDatum rd,ratvec v) = RealForm
compact form of cover

rho_cover

L36rho_cover(RootDatum rd)

dirac_index_ds 2 overloads

L42dirac_index_ds(Param p,KGBElt x_K) = ParamPol
DS with HC parameter(lambda)
K is defined by x_K, then K_cover;
Dirac index is the K_cover-type with infinitesimal character (for K_cover) lambda
L53dirac_index_ds(Param p) = ParamPol
default x_K=KGB(G,0)

dirac_index_std 2 overloads

L56dirac_index_std(ParamPol P,KGBElt x_K) = ParamPol
Dirac index of ParamPol of standards is a linear function
L71dirac_index_std(ParamPol P) = ParamPol

dirac_index_irr 2 overloads

L73dirac_index_irr(Param p,KGBElt x_K) = ParamPol
L75dirac_index_irr(Param p) = ParamPol

show_dirac_index_param_pol

L78show_dirac_index_param_pol(ParamPol P) = void
nice output of Diract index of ParamPol of standards

show_dirac_index 2 overloads

L87show_dirac_index(Param p,KGBElt x_K) = void
nice output of Dirac index of irreducible
L92show_dirac_index(Param p) = void
nice output of Dirac index of irreducible, default x_K=KGB(G,0)

Generated from atlas-scripts at commit 7e1b958 (2026-09-17).