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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 [Car72] will be used, except that the constants are defined for right actions rather than left actions [CMT04].

ExtraspecialPairs(R) : RootDtm -> SeqEnum

The sequence of extraspecial pairs of the root datum R (see [Car72, page 58]). That is the sequence [(r_i, s_i)]_i=1^{N - n} where r_i is minimal such that α_{r_i} + α_{s_i}=α_{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 rth root in the root datum R. That is the pair (s, t) where s is minimal such that α_s + α_t=α_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 α_s - pα_r is a root. This is the same as LeftStringLength. The condition α_s≠∓α_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 α_s + qα_r is a root. This is the same as RightStringLength. The condition α_s≠∓α_r must be satisfied.

CartanInteger(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt

The Cartan integer < α_r, α_s^star > 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 α_s≠∓α_r must be satisfied.

LieConstant_epsilon(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt

The constant ε_rs= Sign(N_rs) for the root datum R. The condition α_s≠∓α_r must be satisfied.

LieConstant_M(R, r, s, i) : RootDtm, RngIntElt, RngIntElt, RngIntElt -> RngIntElt

The constant M_rsi=(1/(i!))N_s₀r ... N_(s_{i - 1}r) where α_{s_i} = iα_r + α_s for the root datum R. The condition α_s≠∓α_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 α_s≠∓α_r and α_r + α_s∈Φ must be satisfied.

LieConstant_eta(R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt

The constant
η_rs= ( - 1)^p_rs (ε_{r, s - pr} ... ε_{r, s - r}/ε_{r, s - pr} ... ε_{r, s + (q - p - 1)r})
for the root datum R. The condition α_s≠∓α_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 Constructors for Lie Algebras.

Example `RootDtm_consts (H104E32)`

The code below verifies some standard formulas in the root datum of type F₄:

> 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]);

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

> C := CartanMatrix(R);
> C[2,3] eq CartanInteger(R,2,3);
true

2.: p_rs is the length of the left string through α_s in the direction of α_r.

> LieConstant_p(R,r,s) eq #LeftString(R,r,s);
true

3.: q_rs is the length of the right string through α_s in the direction of α_r.

> LieConstant_q(R,r,s) eq #RightString(R,r,s);
true

4.: < α_s, α_r^star >=p_rs - q_rs.

> CartanInteger(R,s,r) eq
> LieConstant_p(R,r,s) - LieConstant_q(R,r,s);
true

5.: N_rs = ε_rs(p_rs + 1).

> LieConstant_N(R,r,s) eq
> LieConstant_epsilon(R,r,s) * (LieConstant_p(R,r,s) + 1);
true

V2.28, 13 July 2023