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A129001
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Heights of roots in Cartan root systems for exceptional groups: A1, G2, F4, E6, E7, E8.
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0
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1, 3, 2, 2, 3, 4, 2, 1, 2, 2, 3, 2, 1, 2, 2, 3, 4, 3, 2, 1, 2, 3, 4, 6, 5, 4, 3, 2
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OFFSET
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1,2
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COMMENTS
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Roots heights N(i) Helgasson has in his table for the Cartan roots a(i): delta(n)==Sum[N(i)*a(i),{i,1,n}] h(n)=row sum=Sum[N(i),{i,1,n}] What I found was that my dimension ratio: Dimgroup/DimCartan=h(n )+2 which is not in any of my books. Since exponent sum: Dimgroup=Sum[2*m(i)+1,{i,1,n}] That gives a relationship of sorts between the Poincaré polynomials and the Cartan roots systems: Sum[2*m(i)+1,{i,1,n}]/n=Sum[N(i),{i,1,n}]+2 Table[Apply[Plus, a[n]], {n, 1, 6}] {1, 5, 11, 11, 17, 29} A118889: Table[Apply[Plus, a[n]] + 2, {n, 1, 6}] {3, 7, 13, 13, 19, 31}
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REFERENCES
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Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, pp. 460, 476
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LINKS
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FORMULA
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a(1) = {1}; a(2) = {3, 2}; a(3) = {2, 3, 4, 2}; a(4) = {1, 2, 2, 3, 2, 1}; a(5) = {2, 2, 3, 4, 3, 2, 1}; a(6) = {2, 3, 4, 6, 5, 4, 3, 2};
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EXAMPLE
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{1},
{3, 2},
{2, 3, 4, 2},
{1, 2, 2, 3, 2, 1},
{2, 2, 3, 4, 3, 2, 1},
{2, 3, 4, 6, 5, 4, 3, 2}
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MATHEMATICA
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a[1] = {1}; a[2] = {3, 2}; a[3] = {2, 3, 4, 2}; a[4] = {1, 2, 2, 3, 2, 1}; a[5] = {2, 2, 3, 4, 3, 2, 1}; a[6] = {2, 3, 4, 6, 5, 4, 3, 2}; b = Table[a[n], {n, 1, 6}]; Flatten[b]
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CROSSREFS
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KEYWORD
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nonn,fini,full,tabf,uned
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AUTHOR
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STATUS
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approved
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