GAP (NoCK) - Chapter 2: Obstruction for the existence of compact Clifford-Klein form

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2 Obstruction for the existence of compact Clifford-Klein form
2.1 Technical functions
2.1-1 NonCompactDimension
2.1-2 PCoefficients
2.1-3 PCalculate
2.1-4 AllZeroDH

### 2 Obstruction for the existence of compact Clifford-Klein form

In this chapter we describe functions for algorithm from [BJS+].

#### 2.1 Technical functions

##### 2.1-1 NonCompactDimension

`‣ NonCompactDimension`( G ) | ( function )

For a real Lie algebra \(G\) constructed by the function RealFormById (from [DFdG14]), this function returns the non-compact dimension of \(G\) (dimension of a non-compact part in Cartan decomposition of \(G\)).

    gap> G:=RealFormById("E",6,2); # E6(6)
    <Lie algebra of dimension 78 over SqrtField>
    gap> dG:=NonCompactDimension(G);
    42

##### 2.1-2 PCoefficients

`‣ PCoefficients`( type, rank ) | ( function )

Let \(G\) be a compact connected Lie group of the type type and the rank rank. Let \(\Lambda\,P_{G}=\Lambda (y_1,...,y_l)\) be the exterior algebra over the spaces \(P_G\) of the primitive elements in \(H^*(G)\). Denote the degrees as follows \(|y_j|=2p_j-1,j=1,...,l\). This function returns coefficients \(p_1,\ldots,p_l\).

    gap> PCoefficients("D",5);
    [ 2, 4, 6, 8, 5 ]

##### 2.1-3 PCalculate

`‣ PCalculate`( pi, qi ) | ( function )

Here \(pi=\{ p_1,\ldots,p_l\}\) and \(qi=\{ q_1,\ldots,q_m\}\) are sets of coefficients (\(l\geq m\)). This function returns the polynomial: \(P(t)=\prod_{j=m+1}^l(1+t^{2p_j-1})\prod_{i=1}^m(1-t^{2p_i})/(1-t^{2q_i})\).

    gap> PCalculate([4,2,3],[2,2]);
    t^9+t^5+t^4+1

##### 2.1-4 AllZeroDH

`‣ AllZeroDH`( type, rank, id ) | ( function )

Let \(G^C\) be a complex Lie algebra of the type type and the rank rank. Let \(G\) be a real form of \(G^C\) with the index id (see RealFormsInformation,[DFdG14]). This function returns the set of degrees of \(P(t)\) that have zero coefficients over all permutation (see Section 7 in [BJS+]).

    gap> AllZeroDH("F",4,2);
    [ 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27 ]

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