K_highest_weights.at Function References¶
=¶
=:KHighestWeight (x,mu), KHighestWeight (y,tau)->bool Defined in line number 98.Equality of KHighestWeights; (x,mu)=(x’,mu’) iff x=x’, and mu and mu’ differ by (1-theta)X^* or compact Weyl group action.
LKTs¶
LKTs:Param p->[K_Type] Defined in line number 124.List of LKTs (as final tempered limit parameters) of a module given by a parameter.
LKT¶
LKT:Param p->K_Type Defined in line number 127.LKT of module given by a parameter, if it is unique (otherwise error message)
is_split_spherical¶
is_split_spherical:Param p->bool Defined in line number 134.Test if parameter p for G is G-spherical: G must be rel. split, all roots in tau(p).
is_split_spherical¶
is_split_spherical:K_Type p->bool Defined in line number 143.Test if a K-type is G-spherical: must be LKT of G-spherical p.
highest_weight_split_spherical¶
highest_weight_split_spherical:K_Type p->KHighestWeight Defined in line number 147.Highest weight of a G-spherical K-type (necessarily unique).
highest_weight_split_spherical¶
highest_weight_split_spherical:Param p->KHighestWeight Defined in line number 161.Highest weight of LKT of a G-spherical parameter (is unique).
highest_weight_split_spherical¶
highest_weight_split_spherical:K_Type p,KGBElt x_K->KHighestWeight Defined in line number 165.Highest weight of LKT of G-spherical K-type w.r.t. x_K (is unique).
highest_weight_split_spherical¶
highest_weight_split_spherical:Param p,KGBElt x_K->KHighestWeight Defined in line number 169.Highest weight of LKT of a G-spherical parameter w.r.t. x_K (is unique).
highest_weight_one¶
highest_weight_one:K_Type p->KHighestWeight Defined in line number 173.This function returns just ONE KHighestWeight of a K-type (auxiliary function).
highest_weight_one¶
highest_weight_one:K_Type p,KGBElt x_K->KHighestWeight Defined in line number 192.ONE KHighestWeight of a K-type w.r.t. x_K (auxiliary function).
highest_weights¶
highest_weights:K_Type p->[KHighestWeight] Defined in line number 197.All highest weights of a K-type; R-group acts on one KHighestWeight of previous function (number of terms is |R(K)/R(K,mu)|).
highest_weights¶
highest_weights:K_Type p,KGBElt x_K->[KHighestWeight] Defined in line number 201.All highest weights of a K-type w.r.t. x_K.
highest_weights¶
highest_weights:Param p->[KHighestWeight] Defined in line number 205.List of all highest weights of all LKTs of a parameter.
highest_weights¶
highest_weights:Param p,KGBElt x_K->[KHighestWeight] Defined in line number 211.List of all highest weights of all LKTs of a parameter w.r.t. x_K.
highest_weight¶
highest_weight:K_Type p->KHighestWeight Defined in line number 215.Unique highest weight of a K-type (or error if not unique).
highest_weight¶
highest_weight:K_Type p,KGBElt x_K->KHighestWeight Defined in line number 221.Unique highest weight of a K-type w.r.t. x_K (or error if not unique).
highest_weight¶
highest_weight:Param p->KHighestWeight Defined in line number 227.Unique highest weight of (unique) LKT of a parameter (or error if not unique).
centralizer¶
centralizer:KGBElt x,ratvec v->(KGBElt,RootDatum) Defined in line number 243.(Auxiliary function)
find_nci_root¶
find_nci_root:KGBElt x,ratvec tau->int Defined in line number 253.(Auxiliary function)
tworho_K¶
tworho_K:KGBElt x->ratvec Defined in line number 287.Sum of the roots of K as an element of \((X^*)^{\delta}\otimes\mathbb Q\) (this may be half-integral); x must be in the distinguished fiber.
project_on_dominant_cone¶
project_on_dominant_cone:KGBElt x, ratvec mu->(KGBElt,ratvec,ratvec) Defined in line number 296.Vogan algorithm to project KHighestWeight (x,mu) on dominant cone; returns (x’,mu+2rho_K(x)-rho,tau) with tau dominant and x’ corresponding to the new Weyl chamber.
project_on_dominant_cone¶
project_on_dominant_cone:KGBElt x, vec mu->(KGBElt,ratvec,ratvec) Defined in line number 340.Vogan algorithm; previous function in case mu is given as a vec, rather than ratvec.
characters_order_2¶
characters_order_2:KGBElt x->[vec] Defined in line number 355.(Auxiliary function)
all_G_spherical_same_differential¶
all_G_spherical_same_differential:K_Type p->[K_Type] Defined in line number 375.All G-spherical K-types with same differential as given one.
all_G_spherical_same_differential¶
all_G_spherical_same_differential:Param p->[K_Type] Defined in line number 392.All G-spherical K-types with same differential as the LKT of parameter p.
parabolic_LKT¶
parabolic_LKT:KHighestWeight (x,mu)->Parabolic Defined in line number 404.Parabolic attached to KHighestWeight by Vogan algorithm.
make_strongly_dominant¶
make_strongly_dominant:KHighestWeight mu,KGBElt x_Q->KHighestWeight Defined in line number 419.(Auxiliary function)
K_types¶
K_types:KHighestWeight mu_in->[K_Type] Defined in line number 427.All K_types with the same KHighestWeight.
K_type¶
K_type:KHighestWeight(x,mu)->K_Type Defined in line number 464.K_type with given KHighestWeight if unique (otherwise error).
K0_highest_weight¶
K0_highest_weight:KHighestWeight(x,mu)->Param Defined in line number 474.Parameter for (the RealForm K_0) of the K_0-type with highest weight (the restriction of) KHighestWeight mu.
dimension_rat¶
dimension_rat:KHighestWeight mu->rat Defined in line number 483.Dimension of the K_#-type with KHighestWeight mu.
dimension¶
dimension:KHighestWeight mu->int Defined in line number 484.dimension_rat¶
dimension_rat:K_Type p->rat Defined in line number 487.Dimension of a K-type.
dimension¶
dimension:K_Type p->int Defined in line number 492.H_weight¶
H_weight:KGBElt x,vec mu_K->KHighestWeight Defined in line number 499.(Auxiliary function)
fundamental_weights_K_H¶
fundamental_weights_K_H:KGBElt x->[ratvec] Defined in line number 504.(Auxiliary function)
K0_param¶
K0_param:K_Type p,KGBElt x_K->Param Defined in line number 510.ONE K_0-type in the restriction of a K_type to the identity component K_0 of K (auxiliary function).
K0_param¶
K0_param:K_Type p->Param Defined in line number 516.ONE K_0-type in the restriction of a K_type to the identity component K_0 of K (auxiliary function).
K0_params¶
K0_params:Param p,KGBElt x_K->[Param] Defined in line number 524.All K_0-types in the restriction of the LKTs of parameter p K_0.
K0_params¶
K0_params:Param p->[Param] Defined in line number 529.All K_0-types in the restriction of the LKTs of parameter p to K_0.
K0_param¶
K0_param:Param p,KGBElt x_K->Param Defined in line number 533.Unique K_0-type in the restriction of (unique) LKT to K_0 (error if not unique).
K0_param¶
K0_param:Param p->Param Defined in line number 539.Unique K_0-type in the restriction of (unique) LKT to K_0 (error if not unique).
fundamental_weight_coordinates¶
fundamental_weight_coordinates:KHighestWeight (x,mu)->vec Defined in line number 546.(Auxiliary function)
K_highest_weight_from_fundamental_weights¶
K_highest_weight_from_fundamental_weights:KGBElt x,vec tau->KHighestWeight Defined in line number 552.(Auxiliary function)
dimensions¶
dimensions:Param p, KGBElt x_K->[int] Defined in line number 561.List of the dimensions of the K_0-types in the restriction of the LKTs of parameter p.
dimensions¶
dimensions:Param p->[int] Defined in line number 565.List of the dimensions of the K_0-types in the restriction of the LKTs of parameter p.
dimensions¶
dimensions:[Param] B->[[int]] Defined in line number 569.List of the dimensions of the K_0-types in the restriction of the LKTs of a list of parameters.