Constants Associated with Root Data

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 [Carter, 1972] will be used, except that the constants are defined for right actions rather than left actions [Cohen et al., 2004].

ExtraspecialPairs(R): RootDtm → SeqEnum#

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

NumExtraspecialPairs(R): RootDtm → SeqEnum#

The number of extraspecial pairs of the root datum \(R\). 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.

ExtraspecialPair(R, r): RootDtm, RngIntElt → SeqEnum#

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

ExtraspecialSigns(R): RootDtm → []#

Return the sequence of extraspecial signs of the root datum \(R\).

LieConstant\_p(R, r, s)(): RootDtm, RngIntElt, RngIntElt → RngIntElt#

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

LieConstant\_q(R, r, s)(): RootDtm, RngIntElt, RngIntElt → RngIntElt#

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

CartanInteger(R, r, s): RootDtm, RngIntElt, RngIntElt → RngIntElt#

The Cartan integer \(\langle\alpha_r,\alpha_s^\star\rangle\) for the root datum \(R\).

LieConstant\_N(R, r, s)(): RootDtm, RngIntElt, RngIntElt → RngIntElt#

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

LieConstant\_epsilon(R, r, s)(): RootDtm, RngIntElt, RngIntElt → RngIntElt#

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

LieConstant\_M(R, r, s, i)(): RootDtm, RngIntElt, RngIntElt, RngIntElt → RngIntElt#

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

LieConstant\_C(R, i, j, r, s)(): RootDtm, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt#

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

LieConstant\_eta(R, r, s)(): RootDtm, RngIntElt, RngIntElt → RngIntElt#

The constant

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

for the root datum \(R\). The condition \(\alpha_s\ne\pm\alpha_r\) must be satisfied.

StructureConstants(R): RootDtm → RngIntElt#

The Lie algebra structure constants for the reductive Lie algebra with root datum \(R\) in the sparse format described in Section [DRAFT] Constructors for Lie Algebras.

Example: consts#

The code below verifies some standard formulas in the root datum of type \(F_4\):

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

2. \(p_{rs}\) is the length of the left string through \(\alpha_s\) in the direction of \(\alpha_r\).

3. \(q_{rs}\) is the length of the right string through \(\alpha_s\) in the direction of \(\alpha_r\).

  1. \(\langle\alpha_s,\alpha_r^\star\rangle=p_{rs}-q_{rs}\).

  2. \(N_{rs} = \epsilon_{rs}(p_{rs}+1)\).

> 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;