# 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).

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

(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>.