.. _SectRDConst:

.. _constants-root-datum:

Constants Associated with  Root Data
====================================

In this section functions for a number of constants associated with root data
will be described. These constants are needed to define Lie algebras and groups
of Lie type. The notation of :cite:`Carter-small` will be used,
except that the constants are defined for right actions rather than left actions
:cite:`CohenMurrayTaylor`.

.. magma:function:: ExtraspecialPairs(R)
   :input_types: RootDtm
   :output_types: SeqEnum
   :label: ExtraspecialPairs_RootDtm

   The sequence of extraspecial pairs of the root datum :math:`R` (see
   :raw-latex:`\cite[page 58]{Carter-small}`). That is the sequence
   :math:`[(r_i,s_i)]_{i=1}^{N-n}` where :math:`r_i` is minimal such that
   :math:`\alpha_{r_i}+\alpha_{s_i}=\alpha_{i+n}` (:math:`n` is the rank of
   :math:`R` and :math:`N` is the number of positive roots).

.. magma:function:: NumExtraspecialPairs(R)
   :input_types: RootDtm
   :output_types: SeqEnum
   :label: NumExtraspecialPairs_RootDtm

   The number of extraspecial pairs of the root datum :math:`R`. This function
   doesn’t actually compute the extraspecial pairs, thus is much more efficient
   than calling ``#ExtraspecialPairs(R)`` in case extraspecial pairs are not yet
   computed.

.. magma:function:: ExtraspecialPair(R,r)
   :input_types: RootDtm, RngIntElt
   :output_types: SeqEnum
   :label: ExtraspecialPair_RootDtm_RngIntElt

   The extraspecial pair of the :math:`r`\ th root in the root datum :math:`R`.
   That is the pair :math:`(s,t)` where :math:`s` is minimal such that
   :math:`\alpha_{s}+\alpha_{t}=\alpha_{r}`.

.. magma:function:: ExtraspecialSigns(R)
   :input_types: RootDtm
   :output_types: []
   :label: ExtraspecialSigns_RootDtm

   Return the sequence of extraspecial signs of the root datum :math:`R`.

.. magma:function:: LieConstant\_p(R, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The constant :math:`p_{rs}` for the root datum :math:`R`, i.e. the largest
   :math:`p` such that :math:`\alpha_s-p\alpha_r` is a root. This is the same as .
   The condition :math:`\alpha_s\ne\pm\alpha_r` must be satisfied.

.. magma:function:: LieConstant\_q(R, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The constant :math:`q_{rs}` for the root datum :math:`R`, i.e. the largest
   :math:`q` such that :math:`\alpha_s+q\alpha_r` is a root. This is the same as .
   The condition :math:`\alpha_s\ne\pm\alpha_r` must be satisfied.

.. magma:function:: CartanInteger(R, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt
   :output_types: RngIntElt
   :label: CartanInteger_RootDtm_RngIntElt_RngIntElt

   The Cartan integer :math:`\langle\alpha_r,\alpha_s^\star\rangle` for the root
   datum :math:`R`.

.. magma:function:: LieConstant\_N(R, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The Lie algebra structure constant :math:`N_{rs}` for the root datum :math:`R`.
   The condition :math:`\alpha_s\ne\pm\alpha_r` must be satisfied.

.. magma:function:: LieConstant\_epsilon(R, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The constant :math:`\epsilon_{rs}= \hbox{Sign}(N_{rs})` for the root datum
   :math:`R`. The condition :math:`\alpha_s\ne\pm\alpha_r` must be satisfied.

.. magma:function:: LieConstant\_M(R, r, s, i)
   :input_types: RootDtm, RngIntElt, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The constant :math:`M_{rsi}={1\over{i!}}N_{s_0r}\cdots N_{s_{i-1}r}` where
   :math:`\alpha_{s_i} = i\alpha_r+\alpha_s` for the root datum :math:`R`. The
   condition :math:`\alpha_s\ne\pm\alpha_r` must be satisfied.

.. magma:function:: LieConstant\_C(R, i, j, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The Lie group structure constant :math:`C_{ijrs}` for the root datum :math:`R`.
   The conditions :math:`\alpha_s\ne\pm\alpha_r` and
   :math:`\alpha_r+\alpha_s\in\Phi` must be satisfied.

.. magma:function:: LieConstant\_eta(R, r, s)
   :input_types: RootDtm, RngIntElt, RngIntElt
   :output_types: RngIntElt

   The constant

   .. math::

      \eta_{rs}= (-1)^{p_{rs}} {\epsilon_{r,s-pr}\cdots \epsilon_{r,s-r}\over
        \epsilon_{r,s-pr}\cdots \epsilon_{r,s+(q-p-1)r}}

   for the root datum :math:`R`. The condition :math:`\alpha_s\ne\pm\alpha_r` must
   be satisfied.

.. magma:function:: StructureConstants(R)
   :input_types: RootDtm
   :output_types: RngIntElt
   :label: StructureConstants_RootDtm

   The Lie algebra structure constants for the reductive Lie algebra with root
   datum :math:`R` in the sparse format described in
   Section :ref:`SectAlgLieConstr`.

.. magma:example:: Example: consts
   :label: consts

   The code below verifies some standard formulas in the root datum of type
   :math:`F_4`:

   1. Agreement of the Cartan matrix with the Cartan integers.

   2. :math:`p_{rs}` is the length of the left string through :math:`\alpha_s` in
   the direction of :math:`\alpha_r`.

   3. :math:`q_{rs}` is the length of the right string through :math:`\alpha_s` in
   the direction of :math:`\alpha_r`.

   4. :math:`\langle\alpha_s,\alpha_r^\star\rangle=p_{rs}-q_{rs}`.

   5. :math:`N_{rs} = \epsilon_{rs}(p_{rs}+1)`.

   .. code-block:: magma

      > R := RootDatum("F4");
      > N := NumPosRoots(R);
      > r := Random([1..N]);
      > s := Random([1..r-1] cat [r+1..r+N-1] cat [r+N+1..2*N]);
      > C := CartanMatrix(R);
      > C[2,3] eq CartanInteger(R,2,3);
      true
      %%a> assert $1;
      > LieConstant_p(R,r,s) eq #LeftString(R,r,s);
      true
      %%a> assert $1;
      > LieConstant_q(R,r,s) eq #RightString(R,r,s);
      true
      %%a> assert $1;
      > CartanInteger(R,s,r) eq 
      > LieConstant_p(R,r,s) - LieConstant_q(R,r,s);
      true
      %%a> assert $1;
      > LieConstant_N(R,r,s) eq
      > LieConstant_epsilon(R,r,s) * (LieConstant_p(R,r,s) + 1);
      true
      %%a> assert $1;