Roots and Weights

Function	Argument(s) -> Result(s)
simple_roots	(RootDatum->mat)
simple_coroots	(RootDatum->mat)
posroots	(RootDatum->mat)
poscoroots	(RootDatum->mat)
roots	(RootDatum->mat)
coroots	(RootDatum->mat)
root_coradical	(RootDatum->mat)
coroot_radical	(RootDatum->mat)
fundamental_weight	(RootDatum,int->ratvec)
fundamental_coweight	(RootDatum,int->ratvec)
nr_of_posroots	(RootDatum->int)
root_index	(RootDatum->int)
coroot_index	(RootDatum,vec->int)
integrality_datum	(RootDatum,ratvec->RootDatum)
integrality_points	(Rootdatum,ratvec->[rat])

simple_roots

(RootDatum->mat): matrix of simple roots in the root datum.

simple_coroots

(RootDatum->mat): matrix of simple coroots in the root datum.

posroots

(RootDatum->mat): matrix of positive roots in the root datum.

poscoroots

(RootDatum->mat): matrix of positive coroots in the root datum.

roots

(RootDatum->mat): set of roots in the root datum (columns of result).

coroots

(RootDatum->mat): set of coroots in the root datum (as columns).

root_coradical

(RootDatum->mat): simple roots and coradical basis.

With respect to simple_roots, add columns for coradical basis generators.

coroot_radical

(RootDatum->mat): simple coroots and radical basis.

With respect to simple_coroots, add columns for radical basis generators.

fundamental_weight

(RootDatum,int->ratvec): fundamental weight number \(i\).

fundamental_coweight

(RootDatum,int->ratvec): fundamental coweight number \(i\).

nr_of_posroots

(RootDatum->int): Number of positive roots in the root datum.

root_index

(RootDatum,vec->int): Index of root with respect to posroots.

coroot_index

(RootDatum,vec->int): Index of coroot with respect to poscoroots.

These functions look up the vector in posroots(rd) or poscoroots(rd), and return the index found shifted down by nr_of_posroots(rd), so that the simple roots start at 0, and negative roots give a negative result. In case the vector is not found at all, the value nr_of_posroots(rd) is returned.

integrality_datum

(RootDatum,ratvec->RootDatum): integral coroots subdatum.

Applied to (rd,gamma), forms the root datum whose coroots form the closed subsystem of the coroots of rd that take an integral value on gamma.

integrality_points

(RootDatum,ratvec->[rat]): fractions with integrality.

The call integrality_points(rd,lambda) returns the increasing list of positive fractions f<=1 so f* lambda has more integrality than generically: for some coroot alphav one has integral and nonzero value <alphav,f*lambda>.