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Subalgebra G12+A21 ↪ E16
75 out of 119
Computations done by the calculator project.

Subalgebra type: G12+A21 (click on type for detailed printout).
Subalgebra is (parabolically) induced from G12 .
Centralizer: T1 (toral part, subscript = dimension).
The semisimple part of the centralizer of the semisimple part of my centralizer: E16
Basis of Cartan of centralizer: 1 vectors: (1, 0, -1, 0, 1, -1)

Elements Cartan subalgebra scaled to act by two by components: G12: (2, 3, 4, 6, 4, 2): 6, (-1, -1, -2, -3, -2, -1): 2, A21: (1, 0, 1, 0, 1, 1): 4
Dimension of subalgebra generated by predefined or computed generators: 17.
Negative simple generators: g−29+g−30+g−31, g35, g−7−g−11
Positive simple generators: g31+g30+g29, g−35, −g11+g7
Cartan symmetric matrix: (2/3−10−120001)
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix): (6−30−320004)
Decomposition of ambient Lie algebra: Vω1+2ω3⊕V2ω3⊕2Vω1+ω3⊕2Vω3⊕Vω2⊕Vω1⊕V0
Primal decomposition of the ambient Lie algebra. This decomposition refines the above decomposition (please note the order is not the same as above). Vω1+ω3+6ψ⊕Vω3+6ψ⊕Vω1+2ω3⊕V2ω3⊕Vω2⊕Vω1⊕V0⊕Vω1+ω3−6ψ⊕Vω3−6ψ
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra. As the centralizer is well-chosen and the centralizer of our subalgebra is non-trivial, we may in addition split highest weight vectors with the same weight over the semisimple part over the centralizer (recall that the centralizer preserves the weights over the subalgebra and in particular acts on the highest weight vectors). Therefore we have chosen our highest weight vectors to be, in addition, weight vectors over the Cartan of the centralizer of the starting subalgebra. Their weight over the sum of the Cartans of the semisimple subalgebra and its centralizer is indicated in the third row. The weights corresponding to the Cartan of the centralizer are again indicated with the letter \omega. As there is no preferred way of chosing a basis of the Cartan of the centralizer (unlike the starting semisimple Lie algebra: there we have a preferred basis induced by the fundamental weights), our centralizer weights are simply given by the constant by which the k^th basis element of the Cartan of the centralizer acts on the highest weight vector. Here, we use the choice for basis of the Cartan of the centralizer given at the start of the page.

Highest vectors of representations (total 9) ; the vectors are over the primal subalgebra.−h6+h5−h3+h1−g20+2g19+g17g2g6+g3−g5+g1g25g22−g11+g7g27
weight0ω1ω2ω3ω3ω1+ω3ω1+ω32ω3ω1+2ω3
weights rel. to Cartan of (centralizer+semisimple s.a.). 0ω1ω2ω3−6ψω3+6ψω1+ω3−6ψω1+ω3+6ψ2ω3ω1+2ω3
Isotypic module decomposition over primal subalgebra (total 9 isotypic components).
Isotypical components + highest weightV0 → (0, 0, 0, 0)Vω1 → (1, 0, 0, 0)Vω2 → (0, 1, 0, 0)Vω3−6ψ → (0, 0, 1, -6)Vω3+6ψ → (0, 0, 1, 6)Vω1+ω3−6ψ → (1, 0, 1, -6)Vω1+ω3+6ψ → (1, 0, 1, 6)V2ω3 → (0, 0, 2, 0)Vω1+2ω3 → (1, 0, 2, 0)
Module label W1W2W3W4W5W6W7W8W9
Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element. Cartan of centralizer component.
−h6+h5−h3+h1
−g20+2g19+g17
g−12+2g−15−g−16
−g31+2g30−g29
−h6+h5+h3−h1
2g−29−4g−30+2g−31
−2g16+4g15+2g12
−2g−17−4g−19+2g−20
Semisimple subalgebra component.
−g2
g36
g20+g19−g17
2g−12−2g−15−2g−16
6g−35
−2g31−2g30−2g29
6h6+12h5+18h4+12h3+6h2+6h1
4h6+8h5+12h4+8h3+6h2+4h1
6g−29+6g−30+6g−31
−12g35
−6g16−6g15+6g12
12g−17−12g−19−12g−20
36g−36
36g−2
g6+g3
−g−1+g−5
−g5+g1
g−3+g−6
g25
g−10
g13
−g32
−g−18
−g6+g3
g28
2g−26
−g−1−g−5
2g21
−2g−33
−2g−14
2g9
2g−22
g22
−g−9
g14
−g33
g−21
g5+g1
−g26
2g−28
g−3−g−6
2g18
2g−32
2g−13
2g10
−2g−25
Semisimple subalgebra component.
g11−g7
h6+h5+h3+h1
2g−7−2g−11
g27
−g−4
g20+g17
g34
g−12+g−16
2g8
g11+g7
−g31+g29
−2g−24
−2g−23
h6+h5−h3−h1
−2g23
2g24
−2g−29+2g−31
−2g−7−2g−11
2g−8
2g16+2g12
4g−34
−2g−17−2g−20
4g4
4g−27
Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above0ω1
−ω1+ω2
2ω1−ω2
0
−2ω1+ω2
ω1−ω2
−ω1		ω2
3ω1−ω2
ω1
−ω1+ω2
−3ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
3ω1−2ω2
ω1−ω2
−ω1
−3ω1+ω2
−ω2		ω3
−ω3		ω3
−ω3		ω1+ω3
−ω1+ω2+ω3
ω1−ω3
2ω1−ω2+ω3
−ω1+ω2−ω3
ω3
2ω1−ω2−ω3
−2ω1+ω2+ω3
−ω3
ω1−ω2+ω3
−2ω1+ω2−ω3
−ω1+ω3
ω1−ω2−ω3
−ω1−ω3		ω1+ω3
−ω1+ω2+ω3
ω1−ω3
2ω1−ω2+ω3
−ω1+ω2−ω3
ω3
2ω1−ω2−ω3
−2ω1+ω2+ω3
−ω3
ω1−ω2+ω3
−2ω1+ω2−ω3
−ω1+ω3
ω1−ω2−ω3
−ω1−ω3		2ω3
0
−2ω3		ω1+2ω3
−ω1+ω2+2ω3
ω1
2ω1−ω2+2ω3
−ω1+ω2
ω1−2ω3
2ω3
2ω1−ω2
−ω1+ω2−2ω3
−2ω1+ω2+2ω3
0
2ω1−ω2−2ω3
ω1−ω2+2ω3
−2ω1+ω2
−2ω3
−ω1+2ω3
ω1−ω2
−2ω1+ω2−2ω3
−ω1
ω1−ω2−2ω3
−ω1−2ω3
Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer0ω1
−ω1+ω2
2ω1−ω2
0
−2ω1+ω2
ω1−ω2
−ω1		ω2
3ω1−ω2
ω1
−ω1+ω2
−3ω1+2ω2
2ω1−ω2
0
0
−2ω1+ω2
3ω1−2ω2
ω1−ω2
−ω1
−3ω1+ω2
−ω2		ω3−6ψ
−ω3−6ψ		ω3+6ψ
−ω3+6ψ		ω1+ω3−6ψ
−ω1+ω2+ω3−6ψ
ω1−ω3−6ψ
2ω1−ω2+ω3−6ψ
−ω1+ω2−ω3−6ψ
ω3−6ψ
2ω1−ω2−ω3−6ψ
−2ω1+ω2+ω3−6ψ
−ω3−6ψ
ω1−ω2+ω3−6ψ
−2ω1+ω2−ω3−6ψ
−ω1+ω3−6ψ
ω1−ω2−ω3−6ψ
−ω1−ω3−6ψ		ω1+ω3+6ψ
−ω1+ω2+ω3+6ψ
ω1−ω3+6ψ
2ω1−ω2+ω3+6ψ
−ω1+ω2−ω3+6ψ
ω3+6ψ
2ω1−ω2−ω3+6ψ
−2ω1+ω2+ω3+6ψ
−ω3+6ψ
ω1−ω2+ω3+6ψ
−2ω1+ω2−ω3+6ψ
−ω1+ω3+6ψ
ω1−ω2−ω3+6ψ
−ω1−ω3+6ψ		2ω3
0
−2ω3		ω1+2ω3
−ω1+ω2+2ω3
ω1
2ω1−ω2+2ω3
−ω1+ω2
ω1−2ω3
2ω3
2ω1−ω2
−ω1+ω2−2ω3
−2ω1+ω2+2ω3
0
2ω1−ω2−2ω3
ω1−ω2+2ω3
−2ω1+ω2
−2ω3
−ω1+2ω3
ω1−ω2
−2ω1+ω2−2ω3
−ω1
ω1−ω2−2ω3
−ω1−2ω3
Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.M0Mω1⊕M2ω1−ω2⊕M−ω1+ω2⊕M0⊕Mω1−ω2⊕M−2ω1+ω2⊕M−ω1M3ω1−ω2⊕Mω2⊕Mω1⊕M2ω1−ω2⊕M3ω1−2ω2⊕M−ω1+ω2⊕2M0⊕Mω1−ω2⊕M−3ω1+2ω2⊕M−2ω1+ω2⊕M−ω1⊕M−ω2⊕M−3ω1+ω2Mω3−6ψ⊕M−ω3−6ψMω3+6ψ⊕M−ω3+6ψMω1+ω3−6ψ⊕M2ω1−ω2+ω3−6ψ⊕M−ω1+ω2+ω3−6ψ⊕Mω3−6ψ⊕Mω1−ω2+ω3−6ψ⊕M−2ω1+ω2+ω3−6ψ⊕M−ω1+ω3−6ψ⊕Mω1−ω3−6ψ⊕M2ω1−ω2−ω3−6ψ⊕M−ω1+ω2−ω3−6ψ⊕M−ω3−6ψ⊕Mω1−ω2−ω3−6ψ⊕M−2ω1+ω2−ω3−6ψ⊕M−ω1−ω3−6ψMω1+ω3+6ψ⊕M2ω1−ω2+ω3+6ψ⊕M−ω1+ω2+ω3+6ψ⊕Mω3+6ψ⊕Mω1−ω2+ω3+6ψ⊕M−2ω1+ω2+ω3+6ψ⊕M−ω1+ω3+6ψ⊕Mω1−ω3+6ψ⊕M2ω1−ω2−ω3+6ψ⊕M−ω1+ω2−ω3+6ψ⊕M−ω3+6ψ⊕Mω1−ω2−ω3+6ψ⊕M−2ω1+ω2−ω3+6ψ⊕M−ω1−ω3+6ψM2ω3⊕M0⊕M−2ω3Mω1+2ω3⊕M2ω1−ω2+2ω3⊕M−ω1+ω2+2ω3⊕M2ω3⊕Mω1−ω2+2ω3⊕M−2ω1+ω2+2ω3⊕M−ω1+2ω3⊕Mω1⊕M2ω1−ω2⊕M−ω1+ω2⊕M0⊕Mω1−ω2⊕M−2ω1+ω2⊕M−ω1⊕Mω1−2ω3⊕M2ω1−ω2−2ω3⊕M−ω1+ω2−2ω3⊕M−2ω3⊕Mω1−ω2−2ω3⊕M−2ω1+ω2−2ω3⊕M−ω1−2ω3
Isotypic characterM0Mω1⊕M2ω1−ω2⊕M−ω1+ω2⊕M0⊕Mω1−ω2⊕M−2ω1+ω2⊕M−ω1M3ω1−ω2⊕Mω2⊕Mω1⊕M2ω1−ω2⊕M3ω1−2ω2⊕M−ω1+ω2⊕2M0⊕Mω1−ω2⊕M−3ω1+2ω2⊕M−2ω1+ω2⊕M−ω1⊕M−ω2⊕M−3ω1+ω2Mω3−6ψ⊕M−ω3−6ψMω3+6ψ⊕M−ω3+6ψMω1+ω3−6ψ⊕M2ω1−ω2+ω3−6ψ⊕M−ω1+ω2+ω3−6ψ⊕Mω3−6ψ⊕Mω1−ω2+ω3−6ψ⊕M−2ω1+ω2+ω3−6ψ⊕M−ω1+ω3−6ψ⊕Mω1−ω3−6ψ⊕M2ω1−ω2−ω3−6ψ⊕M−ω1+ω2−ω3−6ψ⊕M−ω3−6ψ⊕Mω1−ω2−ω3−6ψ⊕M−2ω1+ω2−ω3−6ψ⊕M−ω1−ω3−6ψMω1+ω3+6ψ⊕M2ω1−ω2+ω3+6ψ⊕M−ω1+ω2+ω3+6ψ⊕Mω3+6ψ⊕Mω1−ω2+ω3+6ψ⊕M−2ω1+ω2+ω3+6ψ⊕M−ω1+ω3+6ψ⊕Mω1−ω3+6ψ⊕M2ω1−ω2−ω3+6ψ⊕M−ω1+ω2−ω3+6ψ⊕M−ω3+6ψ⊕Mω1−ω2−ω3+6ψ⊕M−2ω1+ω2−ω3+6ψ⊕M−ω1−ω3+6ψM2ω3⊕M0⊕M−2ω3Mω1+2ω3⊕M2ω1−ω2+2ω3⊕M−ω1+ω2+2ω3⊕M2ω3⊕Mω1−ω2+2ω3⊕M−2ω1+ω2+2ω3⊕M−ω1+2ω3⊕Mω1⊕M2ω1−ω2⊕M−ω1+ω2⊕M0⊕Mω1−ω2⊕M−2ω1+ω2⊕M−ω1⊕Mω1−2ω3⊕M2ω1−ω2−2ω3⊕M−ω1+ω2−2ω3⊕M−2ω3⊕Mω1−ω2−2ω3⊕M−2ω1+ω2−2ω3⊕M−ω1−2ω3

Semisimple subalgebra: W_{3}+W_{8}
Centralizer extension: W_{1}

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.
Canvas not supported




Mouse position: (0.00, 0.00)
Selected index: -1
Coordinate center in screen coordinates:
(200.00, 367.50)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00, 0.00)
(0.00, 1.00, 0.00, 0.00)
0: (1.00, 0.00, 0.00, 0.00): (266.67, 467.50)
1: (0.00, 1.00, 0.00, 0.00): (300.00, 567.50)
2: (0.00, 0.00, 1.00, 0.00): (200.00, 367.50)
3: (0.00, 0.00, 0.00, 1.00): (200.00, 367.50)



Made total 1471286 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
3*2 (unknown) gens:
(
x_{1} g_{-29}+x_{2} g_{-30}+x_{3} g_{-31}, x_{9} g_{31}+x_{8} g_{30}+x_{7} g_{29},
x_{4} g_{35}, x_{10} g_{-35},
x_{5} g_{-7}+x_{6} g_{-11}, x_{12} g_{11}+x_{11} g_{7})
h: (2/3, 1, 4/3, 2, 4/3, 2/3), e = combination of g_{29} g_{30} g_{31} , f= combination of g_{-29} g_{-30} g_{-31} h: (-1, -1, -2, -3, -2, -1), e = combination of g_{-35} , f= combination of g_{35} h: (1, 0, 1, 0, 1, 1), e = combination of g_{7} g_{11} , f= combination of g_{-7} g_{-11} Positive weight subsystem: 7 vectors: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (2, 1, 0), (3, 1, 0), (3, 2, 0)
Symmetric Cartan default scale: \begin{pmatrix}
2/3 & -1 & 0\\
-1 & 2 & 0\\
0 & 0 & 2\\
\end{pmatrix}Character ambient Lie algebra: V_{\omega_{1}+2\omega_{3}}+V_{2\omega_{1}-\omega_{2}+2\omega_{3}}+V_{-\omega_{1}+\omega_{2}+2\omega_{3}}+2V_{2\omega_{3}}+V_{\omega_{1}-\omega_{2}+2\omega_{3}}+2V_{\omega_{1}+\omega_{3}}+2V_{2\omega_{1}-\omega_{2}+\omega_{3}}+V_{3\omega_{1}-\omega_{2}}+V_{-2\omega_{1}+\omega_{2}+2\omega_{3}}+V_{-\omega_{1}+2\omega_{3}}+2V_{-\omega_{1}+\omega_{2}+\omega_{3}}+4V_{\omega_{3}}+2V_{\omega_{1}-\omega_{2}+\omega_{3}}+V_{\omega_{2}}+3V_{\omega_{1}}+3V_{2\omega_{1}-\omega_{2}}+V_{3\omega_{1}-2\omega_{2}}+2V_{-2\omega_{1}+\omega_{2}+\omega_{3}}+2V_{-\omega_{1}+\omega_{3}}+3V_{-\omega_{1}+\omega_{2}}+6V_{0}+3V_{\omega_{1}-\omega_{2}}+2V_{\omega_{1}-\omega_{3}}+2V_{2\omega_{1}-\omega_{2}-\omega_{3}}+V_{-3\omega_{1}+2\omega_{2}}+3V_{-2\omega_{1}+\omega_{2}}+3V_{-\omega_{1}}+V_{-\omega_{2}}+2V_{-\omega_{1}+\omega_{2}-\omega_{3}}+4V_{-\omega_{3}}+2V_{\omega_{1}-\omega_{2}-\omega_{3}}+V_{\omega_{1}-2\omega_{3}}+V_{2\omega_{1}-\omega_{2}-2\omega_{3}}+V_{-3\omega_{1}+\omega_{2}}+2V_{-2\omega_{1}+\omega_{2}-\omega_{3}}+2V_{-\omega_{1}-\omega_{3}}+V_{-\omega_{1}+\omega_{2}-2\omega_{3}}+2V_{-2\omega_{3}}+V_{\omega_{1}-\omega_{2}-2\omega_{3}}+V_{-2\omega_{1}+\omega_{2}-2\omega_{3}}+V_{-\omega_{1}-2\omega_{3}}
A necessary system to realize the candidate subalgebra.
x_{2} x_{8} +x_{1} x_{7} -2= 0
x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -3= 0
x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -4= 0
2x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4= 0
x_{3} x_{9} +x_{2} x_{8} -2= 0
x_{3} x_{12} +x_{1} x_{11} = 0
x_{7} x_{12} +x_{9} x_{11} = 0
x_{1} x_{6} +x_{3} x_{5} = 0
x_{4} x_{10} -1= 0
x_{5} x_{11} -1= 0
x_{6} x_{12} -1= 0
x_{6} x_{9} +x_{5} x_{7} = 0
The above system after transformation.
x_{2} x_{8} +x_{1} x_{7} -2= 0
x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -3= 0
x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -4= 0
2x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4= 0
x_{3} x_{9} +x_{2} x_{8} -2= 0
x_{3} x_{12} +x_{1} x_{11} = 0
x_{7} x_{12} +x_{9} x_{11} = 0
x_{1} x_{6} +x_{3} x_{5} = 0
x_{4} x_{10} -1= 0
x_{5} x_{11} -1= 0
x_{6} x_{12} -1= 0
x_{6} x_{9} +x_{5} x_{7} = 0
For the calculator:
(DynkinType =G^{1}_2+A^{2}_1; ElementsCartan =((2/3, 1, 4/3, 2, 4/3, 2/3), (-1, -1, -2, -3, -2, -1), (1, 0, 1, 0, 1, 1)); generators =(x_{1} g_{-29}+x_{2} g_{-30}+x_{3} g_{-31}, x_{9} g_{31}+x_{8} g_{30}+x_{7} g_{29}, x_{4} g_{35}, x_{10} g_{-35}, x_{5} g_{-7}+x_{6} g_{-11}, x_{12} g_{11}+x_{11} g_{7}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{2} x_{8} +x_{1} x_{7} -2, x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -3, x_{3} x_{9} +x_{2} x_{8} +2x_{1} x_{7} -4, 2x_{3} x_{9} +x_{2} x_{8} +x_{1} x_{7} -4, x_{3} x_{9} +x_{2} x_{8} -2, x_{3} x_{12} +x_{1} x_{11} , x_{7} x_{12} +x_{9} x_{11} , x_{1} x_{6} +x_{3} x_{5} , x_{4} x_{10} -1, x_{5} x_{11} -1, x_{6} x_{12} -1, x_{6} x_{9} +x_{5} x_{7} )