The n-Category Café

Hi John,

Take a look at

Maier, Math Comp. 76 (2007) 811-843

http://www.ams.org/mcom/2007-76-258/S0025-5718-06-01939-9/S0025-5718-06-01939-9.pdf

this lists the solutions (see Table 1), the 24 is the order of the group D_3 that permutes these.

Peter wrote:

Maier, The 192 solutions of the Heun equation, Math. Comp. 76 (2007), 811–843.

Thanks very much for this reference, Peter! And thanks for elucidating it, David. This should give me lots of clues as to what’s really going on here.

The $D_n$ Coxeter group seems to be playing a key role. Heun proves:

Theorem 4.3 The automorphism group $G_{asym}(n)$ of asymmetrically reduced $n$-point Fuchsian differential equations is isomorphic to the Coxeter group $D_n$.

I don’t know what all that stuff means, but I can probably figure it out. As far as numerology goes, $D_n$ is is the wreath product of the group of even permutations of $n$ things and the group $\mathbb{Z}_2$, so it has $2^{n-1} n!$ elements. For $n = 3$ this is 24, and for $n = 4$ it’s 192.

Also, someone sent me a helpful email about this, which I’ll take the liberty of quoting:

I guess by now that many people have told you that Kummer’s 24 solutions to the hypergeometric equation are 24 special solutions to the equation, all expressed in terms of the basic hypergeometric function $\,_2F_1(a,b,c;x)$ with various parameter values and multiplication by powers of $x$ and $(1-x)$.

Among solutions to the hypergeometric equation, the $\,_2F_1$ solution is very special. The 23 other solutions that Kummer found are the next most special, since they can all be expressed in terms of hypergeometric series and elementary multipliers. Most solutions to the equation aren’t like this either. The special ones have all sorts of applications, including the computation of the monodromy representation associated to the equation.

In order to understand where Kummer’s solutions come from in a more conceptual way, I recommend

• L. Ehrenpreis, Hypergeometric Functions, in Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday, vol. 1, M. Kashiwara and T. Kawai, eds., pp. 85-128, Academic Press, Boston, MA, 1988.

The whole article is interesting, but the most relevant part is Example 4 on pages 108 and 109. I believe that Ehrenpreis returned to this later, as he promises in the article, but I don’t have the later references handy. He shows that the 24 is the order of the Weyl group of certain torus in a conformal group, which acts naturally on solutions to the hypergeometric equation when it arises by separating variables in a wave equation. He sketches out a very ambitious program, but I don’t think he got very far with it.

The classical way of understanding them is using the Riemann P-function idea. One checks that certain transformations of the independent variable and multiplication by elementary functions give you P-functions with the same singularities and closely related exponents. This is well explained in Whittaker and Watson and also in Poole’s classic book on differential equations. It is more like what is being done in the article by Maier on the arXiv that has already been suggested on your blog — the one about Heun’s equation.

The monodromy calculation is very nicely done in Caratheodory’s complex analysis book (volume 2 maybe), but you can find it in lots of places.

I’ve been maintaining a list of occurences of 24 and its relatives for about 10 years for just such an occasion. (No, I’m not making this up.) By relatives I mean 24, 6 and -1/12 as it’s hard to separate them.

There’s the Bernoulli numbers and various classical formulae involving these things for things like the Riemann zeta function and the Gamma function. There’s the connection with the Riemann-Roch theorem and its generalisations.

There are the countless appearances in the theory of modular forms, Eisenstein series, theta functions, Dedekind eta and so on.

There are connections with the properties of line bundles over moduli spaces of Riemann surfaces.

There are the appearances in combinatorics such as the Golay code, Steiner systems and so on.

There are the connections with group theory, like the Mathieu groups M12 and M24.

There are the virtual Euler numbers of various moduli spaces.

There’s that amazing -1/24 in Rademacher’s formula for the partition function.

Obviously there’s the specialness of 24 for bosonic string theory coming from the conformal anomaly.

24 is the largest n s.t. every nontrivial unit of (Z/nZ)* has order two.

There’s this paper on the number of 12, but it’s really just another occurence of the whole 6/12/24 cluster.

All of the above are intimately related.

There are some nice (related) appearances of 24 in this paper on powers of power series with integer coefficients.

There are the numerous appearances of 24 and friends in the stable homotopy groups of spheres. I’m not sure if this is related but it’s still interesting.

I thought I’d once read that there were 24 exotic differentiable structures on the 4-sphere but I quick check on Wikipedia shows that to be false. I must have made a mistake in my notes. Is there a closely related theorem that I’ve written down incorrectly?

Oh, and I mustn’t forget Todd’s classic paper on the “odd number 6” which leads to constructions for some of the 24 related stuff above.

I think you could squeeze out a whole book on just the number 24.

Watching that 24-cell rotate about a plane is hypnotic. I’ve been reading about it in Sloane, Conway, later edition of Coxeter.

It is especially interesting that in four dimensions the 4-cube is _not_ the only “space-filling” regular polytope. It must share that property with the 4-dimensional cross-polytope (the analog of the octahedron) and with the “24-cell”, so-called because it has 24 octahedral “faces”. And vertices, since, after all, it is self-dual, and has no equivalent in any higher or lower space.

24 vertices of a 24-CELL D4 with squared norm 2:

+1 +1 0 0
+1 0 +1 0
+1 0 0 +1
+1 -1 0 0
+1 0 -1 0
+1 0 0 -1
-1 +1 0 0
-1 0 +1 0
-1 0 0 +1
-1 -1 0 0
-1 0 -1 0
-1 0 0 -1
0 +1 +1 0
0 +1 0 +1
0 +1 -1 0
0 +1 0 -1
0 -1 +1 0
0 -1 0 +1
0 -1 -1 0
0 -1 0 -1
0 0 +1 +1
0 0 +1 -1
0 0 -1 +1
0 0 -1 -1

Fine take the convex hull of those 24 vertices. That gives the 24-cell {3,4,3} with its hypervolume of 8.

Now I apply the Snub operation on it. What is the hypervolume of the polytope? I ask, because I don’t know. Those who do know (Conway, for instance) haven’t replied. Several of us have wrestled with it, including Carl Feynman.

Snub icositetrachoron

Alternative names:
Snub 24-cell
Snub polyoctahedron
Sadi (Jonathan Bowers: for snub
disicositetrachoron)

Symmetry group: [3+,4,3], the ionic diminished icositetrachoric group, of order 576

Schläfli symbols: s{3,4,3}, also s{31,1,1}

Elements:
Cells: 24 icosahedra, 120 tetrahedra
Faces: 480 triangles (96 joining icosahedra to icosahedra, 96 joining tetrahedra to tetrahedra, 288
joining icosahedra to tetrahedra)
Edges: 432
Vertices: 96 (located along each edge of a unit regular icositetrachoron at a distance (sqrt(5)-1)/2 from one of the edge’s ends)

Vertex figure: Tridiminished Icosahedron
Tridiminished icosahedron: a regular icosahedron, edge length 1, from which 3 pentagonal pyramids of triangles are removed and replaced by regular pentagons, yielding a polyhedron whose 8 faces are the 3 pentagons and the 5 remaining equilateral triangles.

While we’re mutating 24-cells, what is the hypervolume of the 144-vertex runcinated 24-cell?

Is there a standard geometric interpretation of hypergeometric functions? It sounds like some of the equations describe the variation of periods over moduli spaces (like Gauss-Manin), but I haven’t seen a general reference about this.

John asked, “Does anyone know what these Kummer solutions are… and why there are 24 of them?”

The following reference to “an aesthetically beautiful result” may be helpful:

“Group Theoretical Aspects of Hypergeometric Functions,” by C. Krattenthaler and K. Srinivasa Rao,
pp. 355-376 in Symmetries in Science

(ed. by Bruno Gruber, Giuseppe Marmo, and Naotaka Yoshinaga– proceedings of “Symmetries in Science XIII,” a symposium held at Mehrerau, Bregenz, Austria, July 20-24, 2003– published by Springer in 2004)

P. 361:

“In [25] it is shown that the 24 complete Kummer solutions (up to a constant) can be related to the permutations which generate the symmetries of the cube and that the mirror symmetries, or the reflection symmetries of the cube, are the ones which correspond to the interchange of the numerator parameters of the ₂F₁ function. The intimate connection between the symmetries of the cube and the 24 Kummer solutions of the Gauss differential equation is an aesthetically beautiful result discovered 90 years after the discovery of the Gauss equation and 66 years after Kummer established its complete set of solutions.”

References, p. 375:

[25] (Expanded from the book’s version)–

“The Finite Group of the Kummer solutions,” by S. Lievens, K. Srinivasa Rao and J. Van der Jeugt, in Integral Transforms and Special Functions, Vol. 16, Issue 2, March 2005, pp. 153-158–

“Abstract: In this short communication, which is self-contained, we show that the set of 24 Kummer solutions of the classical hypergeometric differential equation has an elegant, simple group theoretic structure associated with the symmetries of a cube; or in other words, that the underlying symmetry group is the symmetric group S₄.”

This is not the only discussion of the role of S₄ in the Kummer solutions.

Another paper, “Generating Kummer Type Formulas for Hypergeometric Functions,” by Nobuki Takayama in 2002, discusses “a natural generalization of the method to derive 24 = 4! solutions of the Gauss hypergeometric equation by Kummer.”

There is another place where the number 24 appears which is connected to these in an interesting way. Hyperdeterminants are generalisations of determinants to higher dimensional arrays. They were discovered by Cayley in 1845. The simplest and best known hyperdeterminant is the degree-4 polynomial in the eight variables of the 2x2x2 array, see it here:

http://mathworld.wolfram.com/Hyperdeterminant.html

However, the next one up is the one of real interest here. The hyperdeterminant of a 2x2x2x2 array is a polynomial of degree 24 in its sixteen variables. It has far too many terms to write down but see this link for some details

http://bio.math.berkeley.edu/4cube/

The 2x2x2x2 array can be used to construct elliptic curves. You just contract one of its indices over a 2-vector (x,1) and take the hyperdeterminant of the remaining 2x2x2 array. This gives a quartic in x with coefficients constructed from the elements of the original 2x2x2x2 array so this quartic can be associated with an elliptic curve. The discriminant of this elliptic curve turns out to be our 2x2x2x2 hyperdeterminant so it can be connected to the Ramanujan discriminant function which is the 24th power of the Dedekind eta function. Because the hyperdeterminant is degree 24 it is a weight 12 relative-invariant under GL(2) transformations acting on the 2x2x2x2 array, just as the discriminant function is a weight-12 modular form. So this connects the degree of the hyperdeterminant with the appearance of the number 24 in relation to modular forms.

Because this hyperdeterminant is based on a 4-dimensional cube it might be linked to other ways that the number 24 is connected to 4-dimensional mathematics. For example the 24-cell can be formed from the points at the face centres of the 2x2x2x2 cube as shown by Jonathan Vos Post above.

Finally, I know that another one of John’s favourite numbers is 11 so I can’t resist mentioning that the Newton Polytope of the 2x2x2x2 hyperdeteriminant is 11-dimensional :)

The Lehmers (1974) found an application of the Fano plane for factoring integers via quadratic forms. Here, the triples of forms used form the lines of the projective geometry on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24.

Weisstein, Eric W. “Fano Plane.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FanoPlane.html

The Fano plane is also connected to hyperdeterminants through the Cartan quartic invariant on the 56 dimensional fundamental representation of E_7 which reduces to a 2x2x2 hyperdeterminant when selecting any 3 out of the seven points using the Fano plane or steiner triple system S(2,3,7). It has been applied to physics in these two papers

http://arxiv.org/abs/hep-th/0610314

http://arxiv.org/abs/quant-ph/0609227

I was wondering if there could be a higher order analogue of this for the 2x2x2x2 hyperdeterminant. It might be a degree 24 invariant on the 248 dimensional rep of E_8 that reduces to the hyperdeterminant using the steiner quadruple system S(3,4,8).