Atlas of Lie Groups and Representations

Documentation contents

Script File

hodge_tensor.at

see hodgeTempered.pdf in Dropbox

compute hodge function f\otimes F
f is an arbitrary Hodge function
F is a finite dimensional representation of K,
 given as a function from KGB to [ratvec] (should be [vec]?)
  see tensor_product.at
formula is:
(f\otimes F)(mu)=

NOTE: most of these functions have local versions inside
the definition of hodge_K_type_formula in hodge_K_type_formula.at
Source
atlas-scripts/hodge_tensor.at (129 lines)
Definitions
20
Loads
Loaded by
none of the other all.at files

Definitions in source order

branch_function_std

L21branch_function_std (Param p) = multiplicity_function
branching to K as a function:

hodge_function_std

L26hodge_function_std (Param p) = hodge_function
hodge function on a standard module, this calls
hodge_branch_std, which calls hodge_K_type_formula...

also defined in hodge_K_type_formula.at

shift

L32shift (vec v,hodge_function h) = hodge_function
shift a parameter by adding v to lambda, result may not be standard

*

L35* (i_poly P, hodge_function f) = hodge_function

also defined in basic.at, extParamPol.at, complex.at, modules.at, tits_centralizer.at, stable.at

dual

L42dual((KGBElt ->[ratvec]) weights) = (KGBElt->[ratvec])
f=hodge_function, weights (weight function of a K-type), mu
(f\otimes weights)(mu)=\sum_tau mult(tau,mu\otimes(weights)^*)*f(tau)
this is a finite sum over the K-types tau appearing in mu\otimes(weights)^*

also defined in hodge_K_type_formula.at

hodge_tensor 2 overloads

L45hodge_tensor (hodge_function f, (KGBElt -> [vec]) weights,KType mu) = i_poly
general hodge tensor product
L51hodge_tensor(hodge_function f, (KGBElt -> [vec]) weights) = hodge_function
general hodge tensor product, as a hodge_function

hodge_tensor_wedge_k_u_cap_s

L58hodge_tensor_wedge_k_u_cap_s(hodge_function f,Parabolic P_1,int k) = hodge_function
L\subset L_1\subset G (see hodgeTempered.pdf section 7 or so
hodge tensor product of hodge function on L with wedge^k(u\cap s_1)
u_1 is nilradical of q_1=l_1+u_1, is contained in nilradical u of q=l+u

hodge_tensor_exterior_u_cap_s

L63hodge_tensor_exterior_u_cap_s(hodge_function f,Parabolic P) = hodge_function
same, as a hodge function

hodge_tensor_std 4 overloads

L69hodge_tensor_std(Param p, (KGBElt -> [vec]) weights,KType mu) = i_poly
hodge tensor of hodge_function_std(p) with arbitrary weight function
L74hodge_tensor_std(Param p, (KGBElt -> [vec]) weights) = hodge_function
hodge tensor of hodge_function_std(p) with arbitrary weight function,
as a hodge_function
L78hodge_tensor_std(ParamPol P, (KGBElt -> [vec]) weights,KType mu) = i_poly
hodge tensor of hodge_function_std(ParamPol P) with arbitrary weight function
L84hodge_tensor_std(ParamPol P, (KGBElt -> [vec]) weights) = hodge_function
hodge tensor of hodge_function_std(ParamPol P) with arbitrary weight function,
as a hodge_function

also in this file at line 118

hodge_tensor_wedge_k_u_cap_s_std

L88hodge_tensor_wedge_k_u_cap_s_std(Param p,Parabolic P_1,int k) = hodge_function
hodge tensor hodge_function_std(p) with wedge^k(u_1\cap s)

also in this file at line 121

hodge_tensor_exterior_u_cap_s_std

L97hodge_tensor_exterior_u_cap_s_std(Param p,Parabolic P) = hodge_function
hodge tensor hodge_function_std(p) with wedge^k(u_1\cap s), as a hodge function

also in this file at line 125

Phi_S_inverse

L104Phi_S_inverse (HodgeKTypePol KP, [KType] S) = hodgeParamPol
assume S is sorted by height
map from HodgeKTypePols to hodgeParamPols, given a set S of K-types

Omega_S_inverse

L115Omega_S_inverse(hodge_function f,[KType] S) = hodgeParamPol
map from hodge functions to hodgeParamPols, given a set S of K-types

hodge_tensor_std

L118hodge_tensor_std(Param p, (KGBElt -> [vec]) weights) = hodge_function

also in this file at line 69, line 74, line 78, line 84

hodge_tensor_wedge_k_u_cap_s_std

L121hodge_tensor_wedge_k_u_cap_s_std(Param p,Parabolic P_1,int k) = hodge_function

also in this file at line 88

hodge_tensor_exterior_u_cap_s_std

L125hodge_tensor_exterior_u_cap_s_std(Param p,Parabolic P) = hodge_function

also in this file at line 97

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