Roots and Weights ---------------------- .. list-table:: :widths: 10 20 :header-rows: 1 * - Function - Argument(s) -> Result(s) * - :ref:`simple_roots` - ``(RootDatum->mat)`` * - :ref:`simple_coroots` - ``(RootDatum->mat)`` * - :ref:`posroots` - ``(RootDatum->mat)`` * - :ref:`poscoroots` - ``(RootDatum->mat)`` * - :ref:`roots` - ``(RootDatum->mat)`` * - :ref:`coroots` - ``(RootDatum->mat)`` * - :ref:`root_coradical` - ``(RootDatum->mat)`` * - :ref:`coroot_radical` - ``(RootDatum->mat)`` * - :ref:`fundamental_weight` - ``(RootDatum,int->ratvec)`` * - :ref:`fundamental_coweight` - ``(RootDatum,int->ratvec)`` * - :ref:`nr_of_posroots` - ``(RootDatum->int)`` * - :ref:`root_index` - ``(RootDatum->int)`` * - :ref:`coroot_index` - ``(RootDatum,vec->int)`` * - :ref:`integrality_datum` - ``(RootDatum,ratvec->RootDatum)`` * - :ref:`integrality_points` - ``(Rootdatum,ratvec->[rat])`` .. _simple_roots: simple_roots ++++++++++++++++++ ``(RootDatum->mat)``: matrix of simple roots in the root datum. .. _simple_coroots: simple_coroots +++++++++++++++++++ ``(RootDatum->mat)``: matrix of simple coroots in the root datum. .. _posroots: posroots ++++++++++++ ``(RootDatum->mat)``: matrix of positive roots in the root datum. .. _poscoroots: poscoroots ++++++++++++ ``(RootDatum->mat)``: matrix of positive coroots in the root datum. .. _roots: roots +++++++ ``(RootDatum->mat)``: set of roots in the root datum (columns of result). .. _coroots: coroots ++++++++++ ``(RootDatum->mat)``: set of coroots in the root datum (as columns). .. _root_coradical: root_coradical +++++++++++++++ ``(RootDatum->mat)``: simple roots and coradical basis. With respect to simple_roots, add columns for coradical basis generators. .. _coroot_radical: coroot_radical +++++++++++++++++++ ``(RootDatum->mat)``: simple coroots and radical basis. With respect to ``simple_coroots``, add columns for radical basis generators. .. _fundamental_weight: fundamental_weight ++++++++++++++++++++ ``(RootDatum,int->ratvec)``: fundamental weight number :math:`i`. .. _fundamental_coweight: fundamental_coweight ++++++++++++++++++++++++ ``(RootDatum,int->ratvec)``: fundamental coweight number :math:`i`. .. _nr_of_posroots: nr_of_posroots +++++++++++++++++ ``(RootDatum->int)``: Number of positive roots in the root datum. .. _root_index: root_index +++++++++++ ``(RootDatum,vec->int)``: Index of root with respect to posroots. .. _coroot_index: 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: 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: 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 ````.