A005763 -id:A005763 - cp's OEIS frontend

A129766 Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776.

1, 1, 5, 1, 5, 7, 11, 1, 4, 5, 7, 8, 11, 1, 5, 7, 9, 11, 13, 17, 1, 7, 11, 13, 17, 19, 23, 29

Offset: 1

Roger L. Bagula, May 16 2007

Extra condition of group dimension: b[n] = a[n] + 1 ; DimGroup = Apply[Plus, b[n]]; Table[Apply[Plus, b[n]], {n, 0, 5}] {3, 14, 52, 78, 133, 248} Extra condition of Betti sum: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1,Length[a[i]]}]], t]], {i, 0, 5}] {2, 4, 16, 64, 128, 256} These exponents are necessary to the Poincaré polynomials for these exceptional groups.

			1;
1,5;
1,5,7,11;
1,4,5,7,8,11;
1,5,7,9,11,13,17;
1,7,11,13,17,19,23,29;

Armand Borel, Essays in History of Lie Groups and Algebraic Groups gives G2 Poincaré polynomial, History of Mathematics, V. 21.

Cf. A005556, A005763, A005776.

a[0] = {1}; a[1] = {1, 5}; a[2] = {1, 5, 7, 11}; a[3] = {1, 4, 5, 7, 8, 11}; a[4] = {1, 5, 7, 9, 11, 13, 17}; a[5] = {1, 7, 11, 13, 17, 19, 23, 29}; Flatten[Table[a[n], {n, 0, 5}]]

A005776 Exponents m_i associated with Weyl group W(E_8).

Original entry on oeis.org

1, 7, 11, 13, 17, 19, 23, 29

Offset: 1

N. J. A. Sloane

Numbers coprime to 30 in that number's reduced residue system. - Alonso del Arte, Oct 03 2017

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.

Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018.
Index entries for sequences related to groups.

Cf. A005556 (E_6), A005763 (E_7), A106373, A106374, A106403, A048597.

Exponents(RootDatum("E8")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

Select[Range[30], GCD[30, #] == 1 &] (* Alonso del Arte, Oct 03 2017 *)

select(n->gcd(n,30)==1, [1..29]) \\ Charles R Greathouse IV, Oct 17 2017

A106403 Primitive exponents of the Weyl group W(E_8).

Original entry on oeis.org

3, 15, 23, 27, 35, 39, 47, 59

Offset: 1

N. J. A. Sloane, May 30 2005

C. Chevalley, The Betti numbers of the exceptional Lie groups, pp. 21-24 of Proc. Internat Congress Math., Cambridge 1950, Amer. Math. Soc., 1952.

Cf. A106373, A106374, A106403, A005556, A005763.

Equals 2*A005776(n) + 1.

A089011 a(n) = 1 if n is an exponent of the Weyl group W(E_7), 0 otherwise.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Paul Boddington, Nov 03 2003

The exponents are 1, 5, 7, 9, 11, 13, 17. The point of this sequence is that a similar generating function gives the exponents for any finite Coxeter group.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A005763, A089010.

{a(n)=if(n<1, 0, polcoeff( x^17+x^13+x^11+x^9+x^7+x^5+x, n))} /* Michael Somos, Mar 07 2007 */

Euler transform of length 14 sequence [ 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1]. - Michael Somos, Mar 07 2007

G.f.: x*(1-x^12)*(1-x^14)/((1-x^4)*(1-x^6)).

A106373 Primitive exponents of the Weyl group W(E_6).

Original entry on oeis.org

3, 9, 11, 15, 17, 23

Offset: 1

N. J. A. Sloane, May 30 2005

C. Chevalley, The Betti numbers of the exceptional Lie groups, pp. 21-24 of Proc. Internat Congress Math., Cambridge 1950, Amer. Math. Soc., 1952.

Cf. A106374, A106403, A005556, A005763, A005776.

Equals 2*A005556(n) + 1.

A106374 Primitive exponents of the Weyl group W(E_7).

Original entry on oeis.org

3, 11, 15, 19, 23, 27, 35

Offset: 1

N. J. A. Sloane, May 30 2005

C. Chevalley, The Betti numbers of the exceptional Lie groups, pp. 21-24 of Proc. Internat Congress Math., Cambridge 1950, Amer. Math. Soc., 1952.

Cf. A106373, A106403, A005556, A005763, A005776.

Equals 2*A005763(n) + 1.

A118889 Ratio of Dimensions of the traditional Cartan exceptional group sequence A1,G2,F4,E6,E7,E8 to the Cartan matrix Dimension: Dimc={1, 2, 4, 6, 7, 8} DimG={3, 14, 52, 78, 133, 248} DimG/DimC={3, 7, 13, 13, 19, 31}.

Original entry on oeis.org

3, 7, 13, 13, 19, 31

Offset: 1

Roger L. Bagula, May 17 2007

The sequence is inherently unordered, because there is no standard ordering of these groups. - R. J. Mathar, Dec 04 2011

Cf. A117133, A129766, A005556, A005763, A005776.

(* Cartan Matrices: *)
e[3] = {{2}};
e[4] = {{2, -3}, {-1, 2}};
e[5] = {{2, -1, 0, 0}, {-1, 2, -2, 0}, {0, -1, 2, -1}, {0, 0, -1, 2}};
e[6] = {{2, 0, -1, 0, 0, 0}, {0, 2, 0, -1, 0, 0}, {-1, 0, 2, -1, 0, 0}, { 0, -1, -1, 2, -1, 0}, { 0, 0, 0, -1, 2, -1}, { 0, 0, 0, 0, -1, 2}};
e[7] = {{2, 0, -1, 0, 0, 0, 0}, {0, 2, 0, -1, 0, 0, 0}, {-1, 0, 2, -1, 0, 0, 0}, {0, -1, -1, 2, -1, 0, 0}, {0, 0, 0, -1, 2, -1, 0}, { 0, 0, 0, 0, -1, 2, -1 }, { 0, 0, 0, 0, 0, -1, 2 }};
e[8] = { {2, 0, -1, 0, 0, 0, 0, 0}, { 0, 2, 0, -1, 0, 0, 0, 0}, {-1, 0, 2, -1, 0, 0, 0, 0}, {0, -1, -1, 2, -1, 0, 0, 0}, {0, 0, 0, -1, 2, -1, 0, 0}, { 0, 0, 0, 0, -1, 2, -1, 0}, { 0, 0, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, -1, 2}} ;
a0 = Table[Length[CoefficientList[CharacteristicPolynomial[e[n], x], x]] - 1, {n, 3, 8}]; (* Poincaré Polynomials*)
(*Poincaré polynomial exponents for G2, E6, E7, E8 from A005556, A005763, A005776 and Armand Borel's Essays in History of Lie Groups and Algebraic Groups*) (* b[n] = a[n] + 1 : DimGroup = Apply[Plus, b[n]]*)
a[0] = {1};
a[1] = {1, 5};
a[2] = {1, 5, 7, 11};
a[3] = {1, 4, 5, 7, 8, 11};
a[4] = {1, 5, 7, 9, 11, 13, 17};
a[5] = {1, 7, 11, 13, 17, 19, 23, 29};
b0 = Table[Length[CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]] - 1, {i, 0, 5}];
Table[b0[[n]]/a0[[n]], {n, 1, Length[a0]}]

P[n]=Poincare-Polynomial[n]=Product[1+t^A129766[m],{m,1,n}]
DimG[n]=Length[CoefficientList[P[n],t]]-1
Pc[n]=CharacteristicPolynomial[M[n],x]
DimC[n]=Length[CoefficientList[Pc[n],x]]-1
a[n]=DimG[n]/DimC[n]

A129769 Exponents m(i) for exceptional groups with best guesses for E7 1/2 and E9 added (there is a problem with the dimension of E9 as no sum of odd numbers will equal the 484, I get 483): triangular sequence is: A1,G2,F4,E6,E7 E7 1/2,E8,E9.

Original entry on oeis.org

1, 1, 5, 1, 5, 7, 11, 1, 4, 5, 7, 8, 11, 1, 5, 7, 9, 11, 13, 17, 1, 6, 9, 11, 13, 15, 17, 19, 1, 7, 11, 13, 17, 19, 23, 29, 1, 11, 17, 19, 23, 29, 31, 51, 55

Offset: 1

Roger L. Bagula, May 16 2007

Betti number row sums: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]], {i, 0, 7}] {2, 4, 16, 64, 128, 256, 256, 512} Group dimensions sums: b[n_] = 2*a[n] + 1 Table[Apply[Plus, b[n]], {n, 0, 7}] {3, 14, 52, 78, 133, 190, 248, 483}.

From these exponents it is possible to get Poincaré polynomial estimates for the new E7 1/2 and E8 that best fit the pattern of the known exponents.

J. M. Landsberg, The sextonions and E_{7 1/2} (with L.Manivel) (Advances in Math 201(2006) p143 - 179) page 22; J. M. Landsberg, http://www.math.tamu.edu/~jml/LMsexpub.pdf: The sextonions and E_{7 1/2}
Armand Borel's Essays in History of Lie Groups and Algebraic Groups: gives G2 Poincaré polynomial, History of Mathematics, V. 21; http://www.amazon.com/Essays-History-Groups-Algebraic-Mathematics/dp/0821802887/ref=pd_rhf_p_3/104-0029617-0633535

Cf. A005556, A005763, A005776.

a[0] = {1}; a[1] = {1, 5}; a[2] = {1, 5, 7, 11}; a[3] = {1, 4, 5, 7, 8, 11}; a[4] = {1, 5, 7, 9, 11, 13, 17}; a[5] = {1, 6, 9, 11, 13, 15, 17, 19}; a[6] = {1, 7, 11, 13, 17, 19, 23, 29}; a[7] = {1, 11, 17, 19, 23, 29, 31, 51, 55};

a(0) = {1}; a(1) = {1, 5}; a(2) = {1, 5, 7, 11}; a(3) = {1, 4, 5, 7, 8, 11}; a(4) = {1, 5, 7, 9, 11, 13, 17}; a(5) = {1, 6, 9, 11, 13, 15, 17, 19}; a(6) = {1, 7, 11, 13, 17, 19, 23, 29}; a(7) = {1, 11, 17, 19, 23, 29, 31, 51, 55};

cp's OEIS Frontend

A129766 Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776.

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A005776 Exponents m_i associated with Weyl group W(E_8).

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A106403 Primitive exponents of the Weyl group W(E_8).

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A089011 a(n) = 1 if n is an exponent of the Weyl group W(E_7), 0 otherwise.

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A106373 Primitive exponents of the Weyl group W(E_6).

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A106374 Primitive exponents of the Weyl group W(E_7).

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A118889 Ratio of Dimensions of the traditional Cartan exceptional group sequence A1,G2,F4,E6,E7,E8 to the Cartan matrix Dimension: Dimc={1, 2, 4, 6, 7, 8} DimG={3, 14, 52, 78, 133, 248} DimG/DimC={3, 7, 13, 13, 19, 31}.

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A129769 Exponents m(i) for exceptional groups with best guesses for E7 1/2 and E9 added (there is a problem with the dimension of E9 as no sum of odd numbers will equal the 484, I get 483): triangular sequence is: A1,G2,F4,E6,E7 E7 1/2,E8,E9.

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