Much ado about 248
M.C. Nucci and P.G.L. Leach∗
Dipartimento di Matematica e Informatica, Università di Perugia,
06123 Perugia, Italy
Abstract
In this note we present three representations of a 248-dimensional Lie
algebra, namely the algebra of Lie point symmetries admitted by a system
of five trivial ordinary differential equations each of order forty-four, that
admitted by a system of seven trivial ordinary differential equations each
of order twenty-eight and that admitted by one trivial ordinary differential
equation of order two hundred and forty-four.
1 Introduction
A system of n ordinary differential equations each of order M > 1,
k = fk(u
j , t), j, k = 1, n, s = 0,M − 1, (1)
has a variable number of Lie point symmetries depending upon the structure of
the functions fk. The maximal dimension D of the algebra of admitted Lie point
symmetries can be obtained by the formulæ [9]
M = 2 =⇒ D = n2 + 4n + 3 (2)
M > 2 =⇒ D = n2 +Mn + 3. (3)
Some explicit numbers are given in Table 1.
Recently the elaboration of the elements of the Lie algebra, E8, of order 248 has
been variously announced [3, 7, 13, 17, 16] in the serious popular media. The au-
thoritative source is the Atlas of Lie Groups and Representations [2] which is funded
by the National Science Foundation through the American Institute of Mathematics
[1]. The results of the E8 computation were announced in a talk at MIT by David
Vogan on Monday, March 19, 2007, and the details may be found at [15]. The Atlas
of Lie Groups and Representations is a project to make available information about
representations of semisimple Lie groups over real and p-adic fields. Of particular
importance is the problem of the unitary dual, ie the classification of all of the ir-
reducible unitary representations of a given Lie group. The goal of the Atlas of Lie
∗permanent address: School of Mathematical Sciences, Westville Campus, University of
KwaZulu-Natal, Durban 4000, Republic of South Africa
http://arxiv.org/abs/0704.0096v1
Table 1: : The maximal dimension of the algebra of admitted Lie point symmetries
for systems of equations of varying order (horizontal) and number (vertical).
M 2 3 4 5 6 7 8 9 10
1 8 7 8 9 10 11 12 13 14
2 15 13 15 17 19 21 23 25 27
3 24 21 24 27 30 33 36 39 42
4 35 31 35 39 43 47 51 55 59
5 48 43 48 53 58 63 68 73 78
6 63 57 63 69 75 81 87 93 99
7 80 73 80 87 94 101 108 115 122
8 99 91 99 107 115 123 131 139 147
9 120 111 120 129 138 147 156 165 174
10 143 133 143 153 163 173 183 193 203
Groups and Representations is to classify the unitary dual of a real Lie group, G,
by computer. A step in this direction is to compute the admissible representations
of G including their Kazhdan-Lusztig-Vogan polynomials. The computation for E8
was an important test of the technology. While the computation is an impressive
achievement, it is only a small step towards the unitary dual and should not be
ranked as important as the original work of Kazhdan, Lusztig, Vogan, Beilinson,
Bernstein et al. (See for example [4, 5, 6, 11, 12, 14, 18, 8].) Nevertheless the result
was regarded as being suitable for a concerted campaign of publicity to heighten
awareness of Mathematics in the community at large:
“Symmetrie ist möglicherweise das erfolgreichste Prinzip der Physik überhaupt” [7].
“Un groupe de chercheurs américains et européens, parmi lesquels on trouve deux
Français, est parvenu à décoder une des structures les plus vastes de l’histoire des
mathématiques” [13].
“It may be that some day this calculation can help physicists to understand the
universe” [17].
“Eighteen mathematicians spent four years and 77 hours of supercomputer compu-
tation to describe this structure” [16].
In this note we demonstrate three representations of a Lie algebra of dimension
248. The two of us spent four hours and 77 seconds of pocket-calculator computation
to describe these three structures.
2 Three simple systems
For D = 248 formula (2) does not have integral solutions and so there is no system
of second-order ordinary differential equations of maximal symmetry possessing a
248-dimensional algebra of its Lie point symmetries1. About formula (3) the factors
of 248-3=245 are 1, 5 and 7 (49 is out of question because 492 > 245). Consequently
1Is this another instance of the intrinsically uniqueness of Classical Mechanics?
possible values of n are 1, 5 and 7. The corresponding values ofM are 244, 44 and 28,
respectively. The systems of maximal symmetry are easily obtained as one simply
puts fk = 0 ∀ k. Thus the systems we construct are the simplest representations of
the equivalence class under point transformation of systems of equations of maximal
symmetry.
Firstly we consider the following system:
k = 0, k = 1, 5. (4)
It is easy to show that this simple system admits a 248-dimensional algebra of its
Lie point symmetries since 52 + 5 · 44 + 3 = 248. The algebra is generated by the
operators
Γ1 = t
2∂t + 43t
i=1 ui∂ui ,
Γ2 = t∂t,
Γ3 = ∂t,
Γi,k = uk∂ui , k = 1, 5, i = 1, 5
Γi+5,s = t
s∂ui, s = 0, 43, i = 1, 5.
Secondly we consider the system
u(28)r = 0, r = 1, 7. (6)
This equally simple system admits a 248-dimensional algebra (72+7 · 28+ 3 = 248)
of its Lie point symmetries generated by
Γ1 = t
2∂t + 27t
j=1 uj∂uj ,
Γ2 = t∂t,
Γ3 = ∂t,
Γj,r = ur∂uj , r = 1, 7, j = 1, 7
Γj+7,n = t
n∂uj , n = 0, 27, j = 1, 7.
Thirdly and finally the scalar equation,
u(244) = 0, (8)
admits a 248-dimensional Lie algebra (12+1 ·244+3 = 248) of its point symmetries
generated by the operators
Γ1 = t
2∂t + 243tu∂u,
Γ2 = t∂t,
Γ3 = ∂t,
Γ4 = u∂u,
Γn+5 = t
n∂u, n = 0, 243.
3 Conclusion
We have demonstrated three representations of Lie algebras of dimension 248 which
is the dimension of E8. Although the algebras we present are not simple, their
method of construction is. The reason for this simplicity is that we used represen-
tations for systems of equations of maximal symmetry. We do not deny that larger
systems, be that in order or number, of less than maximal symmetry could possibly
have an algebra of dimension 248, but even on the assumption that such systems
be linear the complexity of the calculation becomes immense [10] and defeats the
purpose of the present note.
Note that we have used the simplest forms for the generators of the algebras of
the three systems, (4), (6) and (8), for our primary interest is the demonstration of
the existence of the algebras. Normally one would use combinations which reflect
subalgebraic structures. For example in the case of (8) for which the algebra is
obviously sl(250, IR) one would replace Γ2 with Γ̃2 = 2t∂t +243u∂u to underline the
subalgebraic structure {sl(2, IR)⊕ A1} ⊕s 244A1, where Γ1, Γ̃2 and Γ3 constitute a
representation of sl(2, IR), Γ4 reflects the homogeneity of the equation in the depen-
dent variable and the 244-element abelian subalgebra is composed of the solution
symmetries, so called because the coefficient functions are solutions of (8).
Acknowledgements
PGLL thanks the University of Kwazulu-Natal for its continued support.
References
[1] American Institute of Mathematics.
http://aimath.org/E8/
[2] Atlas of Lie Groups and Representations.
http://www.liegroups.org/
[3] BBC Monday, 19 March 2007, 12:28 GMT.
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
[4] Beilinson A (1983) Localization of representations of reductive Lie algebras
Proceedings of the International Congress of Mathematicians, Warsaw 699-710
[5] Beilinson A & Bernstein J (1981) Localisation de g-modules Comptes Rendus
de l’Académie des Sciences de Paris Séries I Mathématiques 292 15-18
[6] Bernstein J (1986) On the Kazhdan-Lusztig conjectures AMS Summer Research
Conference (University of California, Santa Cruz, July 1986)
[7] Der Spiegel, 19 März 2007.
http://www.spiegel.de/wissenschaft/mensch/0,1518,472569,00.html
http://aimath.org/E8/
http://www.liegroups.org/
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
http://www.spiegel.de/wissenschaft/mensch/0
[8] Gelfand S & MacPherson R (1982) Verma modules and Schubert cells: a dic-
tionary in Seminaire d’algebre Paul Dubriel et MP Malliavin (Lecture Notes in
Mathematics 925, Springer Verlag, Berlin–New York) 150
[9] González-Gascón F & González-López A (1983) Symmetries of differential equa-
tions IV Journal of Mathematical Physics 24 2006-2021
[10] Gorringe VM & Leach PGL (1988) Lie point symmetries for systems of second
order linear ordinary differential equations Quæstiones Mathematicæ 11 95-117
[11] Kazhdan D & Lusztig G (1979) Representations of Coxeter groups and Hecke
algebras Inventiones Mathematicæ 53 165184
[12] Kazhdan D & Lusztig G (1980) Schubert varieties and Poincaré duality in
Geometry of the Laplace Operator, (Proceedings of Symposium on Pure Math-
ematics 36, American Mathematical Society) 185203
[13] LEMONDE.FR avec AFP 19.03.07
http://www.lemonde.fr/web/article/0,1-0@2-3244,36-884723@51-
884724,0.html
[14] Lusztig G & Vogan D (1983) Singularities of closures of K-orbits on flag mani-
fold Inventiones Mathematicæ 71 365370
[15] http://www.liegroups.org/AIME8/technicaldetails.html
[16] NEW YORK TIMES 2007/03/20.
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
[17] The Times March 19, 2007.
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
[18] Vogan D (1983) Irreducible characters of semisimple Lie groups III: Proof of
the Kazhdan-Lusztig conjecture in the integral case Inventiones Mathematicæ
71 381417
http://www.lemonde.fr/web/article/0
http://www.liegroups.org/AIM$_$E8/technicaldetails.html
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
	Introduction
	Three simple systems
	Conclusion

  In this note we present three representations of a 248-dimensional Lie
algebra, namely the algebra of Lie point symmetries admitted by a system of
five trivial ordinary differential equations each of order forty-four, that
admitted by a system of seven trivial ordinary differential equations each of
order twenty-eight and that admitted by one trivial ordinary differential
equation of order two hundred and forty-four.

Introduction
A system of n ordinary differential equations each of order M > 1,
k = fk(u
j , t), j, k = 1, n, s = 0,M − 1, (1)
has a variable number of Lie point symmetries depending upon the structure of
the functions fk. The maximal dimension D of the algebra of admitted Lie point
symmetries can be obtained by the formulæ [9]
M = 2 =⇒ D = n2 + 4n + 3 (2)
M > 2 =⇒ D = n2 +Mn + 3. (3)
Some explicit numbers are given in Table 1.
Recently the elaboration of the elements of the Lie algebra, E8, of order 248 has
been variously announced [3, 7, 13, 17, 16] in the serious popular media. The au-
thoritative source is the Atlas of Lie Groups and Representations [2] which is funded
by the National Science Foundation through the American Institute of Mathematics
[1]. The results of the E8 computation were announced in a talk at MIT by David
Vogan on Monday, March 19, 2007, and the details may be found at [15]. The Atlas
of Lie Groups and Representations is a project to make available information about
representations of semisimple Lie groups over real and p-adic fields. Of particular
importance is the problem of the unitary dual, ie the classification of all of the ir-
reducible unitary representations of a given Lie group. The goal of the Atlas of Lie
∗permanent address: School of Mathematical Sciences, Westville Campus, University of
KwaZulu-Natal, Durban 4000, Republic of South Africa
http://arxiv.org/abs/0704.0096v1
Table 1: : The maximal dimension of the algebra of admitted Lie point symmetries
for systems of equations of varying order (horizontal) and number (vertical).
M 2 3 4 5 6 7 8 9 10
1 8 7 8 9 10 11 12 13 14
2 15 13 15 17 19 21 23 25 27
3 24 21 24 27 30 33 36 39 42
4 35 31 35 39 43 47 51 55 59
5 48 43 48 53 58 63 68 73 78
6 63 57 63 69 75 81 87 93 99
7 80 73 80 87 94 101 108 115 122
8 99 91 99 107 115 123 131 139 147
9 120 111 120 129 138 147 156 165 174
10 143 133 143 153 163 173 183 193 203
Groups and Representations is to classify the unitary dual of a real Lie group, G,
by computer. A step in this direction is to compute the admissible representations
of G including their Kazhdan-Lusztig-Vogan polynomials. The computation for E8
was an important test of the technology. While the computation is an impressive
achievement, it is only a small step towards the unitary dual and should not be
ranked as important as the original work of Kazhdan, Lusztig, Vogan, Beilinson,
Bernstein et al. (See for example [4, 5, 6, 11, 12, 14, 18, 8].) Nevertheless the result
was regarded as being suitable for a concerted campaign of publicity to heighten
awareness of Mathematics in the community at large:
“Symmetrie ist möglicherweise das erfolgreichste Prinzip der Physik überhaupt” [7].
“Un groupe de chercheurs américains et européens, parmi lesquels on trouve deux
Français, est parvenu à décoder une des structures les plus vastes de l’histoire des
mathématiques” [13].
“It may be that some day this calculation can help physicists to understand the
universe” [17].
“Eighteen mathematicians spent four years and 77 hours of supercomputer compu-
tation to describe this structure” [16].
In this note we demonstrate three representations of a Lie algebra of dimension
248. The two of us spent four hours and 77 seconds of pocket-calculator computation
to describe these three structures.
2 Three simple systems
For D = 248 formula (2) does not have integral solutions and so there is no system
of second-order ordinary differential equations of maximal symmetry possessing a
248-dimensional algebra of its Lie point symmetries1. About formula (3) the factors
of 248-3=245 are 1, 5 and 7 (49 is out of question because 492 > 245). Consequently
1Is this another instance of the intrinsically uniqueness of Classical Mechanics?
possible values of n are 1, 5 and 7. The corresponding values ofM are 244, 44 and 28,
respectively. The systems of maximal symmetry are easily obtained as one simply
puts fk = 0 ∀ k. Thus the systems we construct are the simplest representations of
the equivalence class under point transformation of systems of equations of maximal
symmetry.
Firstly we consider the following system:
k = 0, k = 1, 5. (4)
It is easy to show that this simple system admits a 248-dimensional algebra of its
Lie point symmetries since 52 + 5 · 44 + 3 = 248. The algebra is generated by the
operators
Γ1 = t
2∂t + 43t
i=1 ui∂ui ,
Γ2 = t∂t,
Γ3 = ∂t,
Γi,k = uk∂ui , k = 1, 5, i = 1, 5
Γi+5,s = t
s∂ui, s = 0, 43, i = 1, 5.
Secondly we consider the system
u(28)r = 0, r = 1, 7. (6)
This equally simple system admits a 248-dimensional algebra (72+7 · 28+ 3 = 248)
of its Lie point symmetries generated by
Γ1 = t
2∂t + 27t
j=1 uj∂uj ,
Γ2 = t∂t,
Γ3 = ∂t,
Γj,r = ur∂uj , r = 1, 7, j = 1, 7
Γj+7,n = t
n∂uj , n = 0, 27, j = 1, 7.
Thirdly and finally the scalar equation,
u(244) = 0, (8)
admits a 248-dimensional Lie algebra (12+1 ·244+3 = 248) of its point symmetries
generated by the operators
Γ1 = t
2∂t + 243tu∂u,
Γ2 = t∂t,
Γ3 = ∂t,
Γ4 = u∂u,
Γn+5 = t
n∂u, n = 0, 243.
3 Conclusion
We have demonstrated three representations of Lie algebras of dimension 248 which
is the dimension of E8. Although the algebras we present are not simple, their
method of construction is. The reason for this simplicity is that we used represen-
tations for systems of equations of maximal symmetry. We do not deny that larger
systems, be that in order or number, of less than maximal symmetry could possibly
have an algebra of dimension 248, but even on the assumption that such systems
be linear the complexity of the calculation becomes immense [10] and defeats the
purpose of the present note.
Note that we have used the simplest forms for the generators of the algebras of
the three systems, (4), (6) and (8), for our primary interest is the demonstration of
the existence of the algebras. Normally one would use combinations which reflect
subalgebraic structures. For example in the case of (8) for which the algebra is
obviously sl(250, IR) one would replace Γ2 with Γ̃2 = 2t∂t +243u∂u to underline the
subalgebraic structure {sl(2, IR)⊕ A1} ⊕s 244A1, where Γ1, Γ̃2 and Γ3 constitute a
representation of sl(2, IR), Γ4 reflects the homogeneity of the equation in the depen-
dent variable and the 244-element abelian subalgebra is composed of the solution
symmetries, so called because the coefficient functions are solutions of (8).
Acknowledgements
PGLL thanks the University of Kwazulu-Natal for its continued support.
References
[1] American Institute of Mathematics.
http://aimath.org/E8/
[2] Atlas of Lie Groups and Representations.
http://www.liegroups.org/
[3] BBC Monday, 19 March 2007, 12:28 GMT.
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
[4] Beilinson A (1983) Localization of representations of reductive Lie algebras
Proceedings of the International Congress of Mathematicians, Warsaw 699-710
[5] Beilinson A & Bernstein J (1981) Localisation de g-modules Comptes Rendus
de l’Académie des Sciences de Paris Séries I Mathématiques 292 15-18
[6] Bernstein J (1986) On the Kazhdan-Lusztig conjectures AMS Summer Research
Conference (University of California, Santa Cruz, July 1986)
[7] Der Spiegel, 19 März 2007.
http://www.spiegel.de/wissenschaft/mensch/0,1518,472569,00.html
http://aimath.org/E8/
http://www.liegroups.org/
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
http://www.spiegel.de/wissenschaft/mensch/0
[8] Gelfand S & MacPherson R (1982) Verma modules and Schubert cells: a dic-
tionary in Seminaire d’algebre Paul Dubriel et MP Malliavin (Lecture Notes in
Mathematics 925, Springer Verlag, Berlin–New York) 150
[9] González-Gascón F & González-López A (1983) Symmetries of differential equa-
tions IV Journal of Mathematical Physics 24 2006-2021
[10] Gorringe VM & Leach PGL (1988) Lie point symmetries for systems of second
order linear ordinary differential equations Quæstiones Mathematicæ 11 95-117
[11] Kazhdan D & Lusztig G (1979) Representations of Coxeter groups and Hecke
algebras Inventiones Mathematicæ 53 165184
[12] Kazhdan D & Lusztig G (1980) Schubert varieties and Poincaré duality in
Geometry of the Laplace Operator, (Proceedings of Symposium on Pure Math-
ematics 36, American Mathematical Society) 185203
[13] LEMONDE.FR avec AFP 19.03.07
http://www.lemonde.fr/web/article/0,1-0@2-3244,36-884723@51-
884724,0.html
[14] Lusztig G & Vogan D (1983) Singularities of closures of K-orbits on flag mani-
fold Inventiones Mathematicæ 71 365370
[15] http://www.liegroups.org/AIME8/technicaldetails.html
[16] NEW YORK TIMES 2007/03/20.
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
[17] The Times March 19, 2007.
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
[18] Vogan D (1983) Irreducible characters of semisimple Lie groups III: Proof of
the Kazhdan-Lusztig conjecture in the integral case Inventiones Mathematicæ
71 381417
http://www.lemonde.fr/web/article/0
http://www.liegroups.org/AIM$_$E8/technicaldetails.html
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
	Introduction
	Three simple systems
	Conclusion
