Monodromy Groups and Compositions of Shabat Polynomials

Last week, the great Naiomi Cameron visited me for a few days to discuss some new directions about Shabat polynomials.  I’ve been horrible about posting on this blog, so now that I’ve been motivated to work on Shabat polynomials again, I figured it’s time for me to write!

As considered in the 1994 paper by Georgii Borisovich Shabat and Alexander Zvonkin entitled Plane trees and algebraic numbers, the rational function

\displaystyle \beta(z) = - \dfrac {4}{531441} \, (z - 1) \, z^3 \, \bigl( 2 \, z^2 + 3 \, z + 9 \bigr)^3 \, \bigl( 8 \, z^4 + 28 \, z^3 + 126 \, z^2 + 189 \, z + 378 \bigr)

is a Shabat polynomial which happens to be the composition \beta = \phi \circ \Phi of two other Shabat polynomials.  The first has monodromy group G_\phi = Z_2 as a cyclic group, while the second has monodromy group G_\phi = A_7 as an alternating group.  The monodromy group of the composition has order |G_\beta| = 12 \, 700 \, 800 = |G_\phi| \cdot |G_\Phi|^2. Do we have G_\beta = Z_2 \ltimes \bigl( A_7 \times A_7 \bigr) as the wreath product of G_\Phi by G_\phi?

Belyi Maps, Degree Sequences, and Passports

Let me begin by establishing some notation and making some remarks.  A lot of this to begin with will be background information, so I apologize to those who already know this stuff pretty well.

Let X be a compact connected Riemann surface. Remember that the only case we care about is when X is the Riemann sphere. Indeed, we have a bijection which comes from stereographic projection:

\displaystyle \begin{matrix} X = \mathbb P^1(\mathbb C) = \mathbb C \cup \{ \infty \} & \longrightarrow & S^2(\mathbb R) = \left \{ (u,v,w) \in \mathbb A^3(\mathbb R) \, \biggl| \, u^2 + v^2 + w^2 = 1 \right \} \\[10pt] z = x + i \, y = \dfrac {u + i \, v}{1 - w} & \mapsto &(u,v,w) = \left( \dfrac {2 \, x}{x^2 + y^2 + 1}, \ \dfrac {2 \, y}{x^2 + y^2 + 1}, \ \dfrac {x^2 + y^2 - 1}{x^2 + y^2 + 1} \right) \end{matrix}

A Belyi map \beta: X \to \mathbb P^1(\mathbb C) is a morphism which is branched only above three points \{ w_0, \, w_1, \, w_\infty \} \subseteq \mathbb P^1(\mathbb C). When X = \mathbb P^1(\mathbb C) is a Riemann sphere, a Belyi map must be the ratio of two polynomials having complex coefficients. Rather implicitly:

\displaystyle \dfrac {w_1 - w_\infty}{w_1 - w_0} \, \dfrac {\beta(z) - w_0}{\beta(z) - w_\infty} = \beta_0 \cdot \dfrac {\prod_{i = 1}^n \bigl(z - z_0^{(i)} \bigr)^{b_i}}{\prod_{k = 1}^p \bigl(z - z_\infty^{(k)} \bigr)^{f_k}} = 1 + \beta_1 \cdot \dfrac {\prod_{j = 1}^m \bigl(z - z_1^{(j)} \bigr)^{w_j}}{\prod_{k = 1}^p \bigl(z - z_\infty^{(k)} \bigr)^{f_k}}.

Once we let e_i be the number of times b_i (w_i, or f_i, respectively) appears in the exponents \{ b_1, \, \dots, \, b_n \} (\{ w_1, \, \dots, \, w_m \}, or \{ f_1, \, \dots, \, f_p \}, respectively), we define the degree sequence \mathcal D of a Belyi map as a multiset of multisets:

\displaystyle \mathcal D = \biggl \{ \bigl \{ b_1, \, \dots, \, b_n \bigr \}, \, \bigl \{ w_1, \, \dots, \, w_m \bigr \}, \, \bigl \{ f_1, \, \dots, \, f_p \bigr \} \biggr \}

and the passport \Pi of a Belyi map as shorthand notation for the degree sequence:

\displaystyle \Pi = \left[ \prod_{i} {b_i}^{e_i}; \ \prod_j {w_j}^{e_j}; \ \prod_k {f_k}^{e_k} \right]

where

\displaystyle \deg(\beta) = \sum_{i = 1}^n b_i = \sum_{i} e_i \, b_i = \sum_{j = 1}^m w_j = \sum_{j} e_j \, w_j = \sum_{k = 1}^p f_k = \sum_{k} e_k \, f_k.

These quantities give combinatorial information about the critical points of \beta, namely, the three sets \beta^{-1}(w_0) = \bigl \{ z_0^{(1)}, \, \dots, \, z_0^{(n)} \bigr \}, \beta^{-1}(w_1) = \bigl \{ z_1^{(1)}, \, \dots, \, z_1^{(m)} \bigr \}, and \beta^{-1}(w_\infty) = \bigl \{ z_\infty^{(1)}, \, \dots, \, z_\infty^{(p)} \bigr \}. For example, they correspond to partitions of the integer N = \deg(\beta) into n, m, and p parts, respectively.

Shabat Polynomials

We say \beta is a Shabat polynomial if \beta is a Belyi map and \beta^{-1}(w_\infty) = \{ z_\infty \} consists of just p = 1 point in the preimage. By pre- and post-composing with Moebius transformations, say

\displaystyle \begin{matrix} \mathbb P^1(\mathbb C) & Z & 0 & 1 & \infty \\[10pt] \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\[10pt] \mathbb P^1(\mathbb C) &z =\dfrac {z_\infty \, (z_0 - z_1) \, Z + z_0 \, (z_1 - z_\infty)}{(z_0 - z_1) \, Z + (z_1 - z_\infty)} & z_0 & z_1 & z_\infty \\[10pt] \downarrow & \downarrow & \downarrow & \downarrow &\downarrow \\[10pt]\mathbb P^1(\mathbb C) & w = \beta(z) & w_0 & w_1 & w_\infty\\[10pt] \downarrow & \downarrow & \downarrow & \downarrow &\downarrow \\[10pt]\mathbb P^1(\mathbb C) &W = \dfrac {w_1 - w_\infty}{w_1 - w_0} \, \dfrac {w - w_0}{w - w_\infty} & 0 & 1 & \infty \end{matrix}

we may as well assume that a Shabat polynomial \beta: X \to \mathbb P^1(\mathbb C) is a Belyi map branched only above \{ 0, \, 1, \, \infty \} such that \beta \bigl( \{ 0, \, 1, \, \infty \} \bigr) \subseteq \{ 0, \, 1, \, \infty \} and \beta^{-1}(\infty) = \{ \infty \}. In particular, we can factor a Shabat polynomial as

\displaystyle \beta(z) = \beta_0 \prod_{i = 1}^n \bigl(z - z_0^{(i)} \bigr)^{b_i} = 1 + \beta_1 \prod_{j = 1}^m \bigl(z - z_1^{(j)} \bigr)^{w_j}.

We often omit the information about infinity and write the degree sequence/passport of a Shabat polynomial as

\displaystyle \begin{matrix} \mathcal D & = & \biggl \{ \bigl \{ b_1,\, b_2, \, \dots, \, b_n \bigr \}, \, \bigl \{ w_1, \, w_2, \, \dots, \, w_m \bigr \} \biggr \} \\[5pt] \Pi & = &\displaystyle \left[ \prod_{i} {b_i}^{e_i}; \ \prod_j {w_j}^{e_j} \right] \end{matrix}

where as before

\displaystyle \deg(\beta) = \sum_{i = 1}^n b_i = \sum_{i} e_i \, b_i = \sum_{j = 1}^m w_j = \sum_{j} e_j \, w_j.

There is a simple way to generate lots of examples of Shabat polynomials. Say that \Phi: X \to \mathbb P^1(\mathbb C) is one example of a Shabat polynomial. It is well-known that we can generate more Shabat polynomials \beta = \phi \circ \Phi : X \to \mathbb P^1(\mathbb C) as the composition of two maps \Phi: X \to \mathbb P^1(\mathbb C) and \phi: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) as long as \phi is a Shabat polynomial which satisfies \phi \bigl( \{ 0, \, 1, \, \infty \} \bigr) \subseteq \{ 0, \, 1, \, \infty \}. One obvious such map is \phi(z) = 4 \, z \, (1-z), although of course there are many others. I will return to this construction later.

Dessins d’Enfants and Monodromy Groups

Given a Belyi map \beta: X \to \mathbb P^1(\mathbb C), its Dessin d’Enfant \Delta_\beta is a bipartite graph which can be embedded on the Riemann sphere X = \mathbb P^1(\mathbb C). It is defined as follows.

  • The “black” vertices are B = \beta^{-1}(w_0) = \bigl \{ z_0^{(1)}, \, \dots, \, z_0^{(n)} \bigr \};
  • The “white” vertices are W = \beta^{-1}(w_1) = \bigl \{ z_1^{(1)}, \, \dots, \, z_1^{(m)} \bigr \};
  • The “midpoints of the faces” are F = \beta^{-1}(w_\infty) = \bigl \{ z_\infty^{(1)}, \, \dots, \, z_\infty^{(p)} \bigr \}; and
  • The “edges” are E = \beta^{-1}\bigl( [0,1] \bigr) as the inverse image of the line segment 0 \leq w \leq 1.

The Dessin d’Enfant will have |B| + |W| = n + m vertices, |E| = \deg(\beta) = N edges, and |F| = p faces, so the Euler characteristic forces N = m+n+p - 2. When \beta is a Shabat polynomial, there is only p = 1 face, so the corresponding Dessin d’Enfant will be a tree.  There is a nice post on MathOverflow which discusses this in a bit more detail.

In addition to Belyi maps/Shabat polynomials \beta and degree sequences \mathcal D/passports \Pi, there is a third object of interest. The monodromy group G_\beta is a transitive subgroup of $S_N$ which is constructed as follows.

  1. Label the N edges of \Delta_\beta from 1 through N.
  2. For each “black” vertex z_0^{(i)} \in B, read off the labels of the b_i adjacent edges going counterclockwise, say as a b_i-cycle \sigma_0^{(i)} = \bigl( B_1 \, B_2 \, \cdots \, B_{b_i} \bigr) in the symmetric group.
  3. For each “white” vertex z_1^{(i)} \in W, read off the labels of the w_j adjacent edges going counterclockwise, say as a w_j-cycle \sigma_1^{(j)} = \bigl( W_1 \, W_2 \, \cdots \, W_{w_j} \bigr) in the symmetric group.
  4. For each “midpoint of edge” z_\infty^{(k)} \in F, read off the labels of the f_k adjacent edges going counterclockwise, say as a f_k-cycle \sigma_\infty^{(k)} = \bigl( F_1 \, F_2 \, \cdots \, F_{f_k} \bigr) in the symmetric group.
  5. Form that subgroup G_\beta = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle of S_N which is generated by the products of disjoint cycles \displaystyle \sigma_0 = \prod_{i = 1}^n \sigma_0^{(i)}, \displaystyle \sigma_1 = \prod_{j = 1}^m \sigma_1^{(j)} and \displaystyle \sigma_\infty = \prod_{k = 1}^p \sigma_\infty^{(k)}.

If \sigma_0', \, \sigma_1', \, \sigma_\infty' is a different set of permutations arising from this construction but from a different labeling of the edges, then there is a permutation \tau \in S_N so that these are simultaneously conjugate to \sigma_0, \, \sigma_1, \, \sigma_\infty, that is, \sigma_0' = \tau \, \sigma_0 \, \tau^{-1}, \sigma_1' = \tau \, \sigma_1 \, \tau^{-1}, and \sigma_\infty' = \tau \, \sigma_\infty \, \tau^{-1}. This permutation \tau must send the edge labels B_i \mapsto B'_i, W_j \mapsto W'_j, and F_k \mapsto F'_k.

It is not too difficult to recover the Dessin d’enfant \Delta_\beta from a triple (\sigma_0, \, \sigma_1, \, \sigma_\infty). Indeed, the number n (m, respectively) of disjoint cycles in the decomposition of \sigma_0 (\sigma_1, respectively) gives the number of “black” (“white”, respectively) vertices, and one can read off the cycle notation to find the labels of the edges which are adjacent to each “black” (“white”, respectively) vertex. One can connect the edges then slide around the vertices as necessary so that the faces match the disjoint cycle decomposition of \sigma_\infty.

Relating Belyi Maps and Monodromy Groups

Fix a positive integer N, and let S_N denote the symmetric group of degree N, and let \mathcal D = \bigl \{ \{ b_1,\, b_2, \, \dots, \, b_n \}, \, \{ w_1, \, w_2, \, \dots, \, w_m \}, \, \{ f_1, \, f_2, \, \dots, \, f_p \} \bigr \} be a multiset of multisets such that N = \sum_i b_i = \sum_j w_j = \sum_k f_k = m + n + p - 2.

Proposition. The following are equivalent:

  • \mathcal D is the degree sequence of a Belyi map \beta: X \to \mathbb P^1(\mathbb C) of degree N on the Riemann sphere X = \mathbb P^1(\mathbb C).
  • There is a triple (\sigma_0, \, \sigma_1, \, \sigma_\infty) of elements in S_N which can be expressed as a disjoint product of cycles \displaystyle \sigma_0 = \prod_{i = 1}^n \bigl( B_1 \, \cdots \, B_{b_i} \bigr), \displaystyle \sigma_1 = \prod_{j = 1}^m \bigl( W_1 \, \cdots \, W_{w_j} \bigr), and \displaystyle \sigma_\infty = \prod_{k = 1}^p \bigl( F_1 \, \cdots \, F_{f_k} \bigr) such that \sigma_0 \, \sigma_1 \, \sigma_\infty = 1 and G = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle is a transitive subgroup of S_N.

The result was established by Adolf Hurwitz in his 1891 paper Ueber Riemann’sche Flachen mit gegebenen Verzweigungspunkten. I’ve explained how to construct a degree sequence from a Belyi map, but of course the opposite direction is an active area of research.

Proposition. All permutations \sigma_0, \, \sigma_0' of cycle type (b_1, \, b_2, \, \dots, \, b_n) are conjugate, that is, \sigma_0' = \tau \, \sigma_0 \, \tau^{-1} for some \tau. In particular, all triples (\sigma_0, \, \sigma_1, \, \sigma_\infty), \, (\sigma_0', \, \sigma_1', \, \sigma_\infty') of “cycle type \mathcal D” are conjugate, that is, \sigma_0' = \tau_0 \, \sigma_0 \, \tau_0^{-1}, \sigma_1' = \tau_1 \, \sigma_1 \, \tau_1^{-1}, and \sigma_\infty' = \tau_\infty \, \sigma_\infty \, \tau_\infty^{-1} for some \tau_0, \, \tau_1, \, \tau_\infty \in S_N.

This is an elementary fact about permutations.

Proposition. There is a one-to-one correspondence between:

  • Belyi maps \beta: X \to \mathbb P^1(\mathbb C) of degree N on some compact connected Riemann surface X.
  • Triples (\sigma_0, \, \sigma_1, \, \sigma_\infty) of elements in S_N, modulo simultaneous conjugation, such that \sigma_0 \, \sigma_1 \, \sigma_\infty = 1 and G = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle is a transitive subgroup of S_N.

Proposition. Say that (\sigma_0, \, \sigma_1, \, \sigma_\infty) is a triple of elements in S_N such that \sigma_0 \, \sigma_1 \, \sigma_\infty = 1, and denote G = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle as the subgroup of S_N generated by them. Denote C_G(\sigma) = \bigl \{ \gamma \in F \, \bigl| \, \gamma \, \sigma \, \gamma^{-1} = \sigma \bigr \} as the centralizer of \sigma in G. There is a one-to-one correspondence between:

  • The set of all triples (\sigma_0', \, \sigma_1', \, \sigma_\infty'), modulo simultaneous conjugation, such that \sigma_0' \, \sigma_1' \, \sigma_\infty' = 1 and \sigma_0' = \tau_0 \, \sigma_0 \, \tau_0^{-1}, \sigma_1' = \tau_1 \, \sigma_1 \, \tau_1^{-1} for some \tau_0, \, \tau_1 \in S_N
  • Double cosets in C_G(\sigma_0) \backslash G \slash C_G(\sigma_1).

These correspondences between Belyi maps, triples (\sigma_0, \, \sigma_1, \, \sigma_\infty), and double cosets C_G(\sigma_0) \backslash G \slash C_G(\sigma_1) were established by Michael Klug, Michael Musty, Sam Schiavone, and John Voight, in their 2013 paper Numerical calculation of three-point branched covers of the projective line. You can find the paper online at http://arxiv.org/abs/1311.2081v3. For the third correspondence, I’ve explained how to construct a triple of permutations from a Belyi map. Here’s a quick sketch of the fourth correspondence: say that (\sigma_0', \, \sigma_1', \, \sigma_\infty') and (\sigma_0'', \, \sigma_1'', \, \sigma_\infty'') are two triples such that

\displaystyle \begin{matrix} \sigma_0' \, \sigma_1' \, \sigma_\infty' = 1 & \qquad \qquad & \sigma_0' = \tau_0 \, \sigma_0 \, \tau_0^{-1} & \qquad \qquad & \sigma_0'' = \theta_0 \, \sigma_0 \, \theta_0^{-1} \\[15pt] \sigma_0'' \, \sigma_1'' \, \sigma_\infty'' = 1 & & \sigma_1' = \tau_1 \, \sigma_1 \, \tau_1^{-1} & & \sigma_1'' = \theta_1 \, \sigma_1 \, \theta_1^{-1} \end{matrix}

for some \tau_0, \, \tau_1, \, \theta_0, \, \theta_1 \in S_N, so denote \eta' = \tau_0^{-1} \, \tau_1 and \eta'' = \theta_0^{-1} \, \theta_1. Assume that they generate the same the double coset C_G(\sigma_0) \, \eta \, C_G(\sigma_1), that is, there exist \gamma_0 \in C_G(\sigma_0) and \gamma_1 \in C_G(\sigma_1) such that \eta'' = \gamma_0 \, \eta' \, \gamma_1. We have the identity

\displaystyle \begin{matrix} \sigma_0'' & = & \tau \, \sigma_0' \, \tau^{-1} \\[5pt] \sigma_1'' & = & \tau \, \sigma_1', \, \tau^{-1} \\[5pt] \sigma_\infty'' & = & \tau \, \sigma_\infty', \, \tau^{-1} \end{matrix}

in terms of \tau = \theta_0 \, \gamma_0 \, \tau_0^{-1} = \theta_1 \, \gamma_1^{-1} \, \tau_1^{-1} so that the two triples are simultaneously conjugate. Conversely, assume that the triples are simultaneously conjugate. Then the elements \gamma_0 = \theta_0^{-1} \, \tau \, \tau_0 \in C_G(\sigma_0) and \gamma_1 = \tau_1^{-1} \, \tau^{-1} \, \theta_1 \in C_G(\sigma_1), so that \eta' and \eta'' = \gamma_0 \, \eta' \, \gamma_1 generate the same the double coset C_G(\sigma_0) \, \eta \, C_G(\sigma_1).

Algorithm: Counting Belyi Maps from Degree Sequences

The monodromy group G_\beta helps to count the number of Belyi maps \beta of degree N. Here’s an algorithm:

  1. Fix a positive integer N.
  2. Compute all multisets of multisets

    \displaystyle \mathcal D = \biggl \{ \bigl \{ b_1,\, b_2, \, \dots, \, b_n \bigr \}, \, \bigl \{ w_1, \, w_2, \, \dots, \, w_m \bigr \}, \, \bigl \{ f_1, \, f_2, \, \dots, \, f_p \bigl \} \biggr \}

     such that N = \sum_i b_i = \sum_j w_j = \sum_k f_k = m + n + p - 2.

  3. For each \mathcal D, find a triple (\sigma_0, \, \sigma_1, \, \sigma_\infty) of elements in S_N which can be expressed as a disjoint product of cycles \displaystyle \sigma_0 = \prod_{i = 1}^n \bigl( B_1 \, B_2 \, \cdots \, B_{b_i} \bigr), \displaystyle  \sigma_1 = \prod_{j = 1}^m \bigl( W_1 \, W_2 \, \cdots \, W_{w_j} \bigr), and \displaystyle \sigma_\infty = \prod_{k = 1}^p \bigl( F_1 \, F_2 \, \cdots \, F_{f_k} \bigr) such that \sigma_0 \, \sigma_1 \, \sigma_\infty = 1 and G = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle is a transitive subgroup of S_N. If no such triple exists, there is no Belyi map for which \mathcal D is a degree sequence.
  4. Compute C_G(\sigma_0) and C_G(\sigma_1) as the centralizers of \sigma_0 and \sigma_1 in G. Given a double-coset representative \eta in C_G(\sigma_0) \backslash G \slash C_G(\sigma_1), the triple corresponding to \sigma_0' = \sigma_0 and \sigma_1' = \eta \, \sigma_1 \, \eta^{-1} gives a unique Belyi map.

Example #1

Let me give a relatively simple example. The rational \beta(z) = z^N is a Shabat polynomial of degree N whose passport is \Pi = \bigl[ N^1; \, 1^N \bigr], Dessin d’Enfant is the Star Graph with N spokes, and whose monodromy group is G_\beta = Z_N as the cyclic group of order N. To see why the latter is true, observe that

  • the “black” vertex is z_0^{(1)} = 0,
  • the “white” vertices are z_1^{(k)} = \cos (2 \pi k/N) + i \, \sin (2 \pi k / N) for k = 1, \, 2, \, \dots, \, N,
  • the “midpoint of the face” is z_\infty^{(1)} = \infty, and
  • the edges are the segments e_k = \left \{ (1-t) \, z_0^{(1)} + t \, z_1^{(k)} \, \biggl| \, 0 \leq t \leq 1 \right \} for k = 1, \, 2, \, \dots, \, N.

If we label edge e_k as the positive integer “k”, then \sigma_0 = (1 \, 2 \, \cdots \, N) is an N-cycle while \sigma_1 = 1 is trivial. Hence the group generated by \sigma_0, \sigma_1, and \sigma_\infty = (\sigma_0 \, \sigma_1)^{-1} is G_\beta = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle = Z_N.

In fact, I can do even better: I claim this is the only Shabat polynomial associated with this passport! Indeed, if we have a triple (\sigma_0, \, \sigma_1, \, \sigma_\infty) associated with the passport \Pi = \bigl[ N^1; \, 1^N \bigr], then \sigma_1 must be the trivial permutation, and so the centralizer C_G(\sigma_1) = G. This means there are no nontrivial double cosets in C_G(\sigma_0) \backslash G \slash C_G(\sigma_1), so there can only be one Shabat polynomial.

Example #2

As considered in the 1994 paper by Georgii Borisovich Shabat and Alexander Zvonkin entitled Plane trees and algebraic numbers, the rational function

\displaystyle \beta(z) = - \dfrac {4}{531441} \, (z - 1) \, z^3 \, \bigl( 2 \, z^2 + 3 \, z + 9 \bigr)^3 \, \bigl( 8 \, z^4 + 28 \, z^3 + 126 \, z^2 + 189 \, z + 378 \bigr)

is a Shabat polynomial of degree N = 14 whose passport is \Pi = \bigl[ 3^3 \, 1^5; \ 2^7 \bigr]. Observe that \beta = \phi \circ \Phi is the composition of the two rational functions \phi(z) = 4 \, z \, (1-z) and \displaystyle \Phi(z) = - \dfrac {1}{729} \, (z - 1) \, (2 \, z^2 + 3 \, z + 9)^3 which are both Shabat polynomials, one of degree \deg(\phi) = 2 with passport \Pi_\phi = \bigl[ 1^2; \ 2^1 \bigr], and the other of degree \deg(\Phi) = 7 with passport \Pi_\Phi = \bigl[ 3^2 \, 1^1; \ 3^1 \, 1^4 \bigr]. Since \Phi(z) = 1 - \bigl( 2 \, z - 1 \bigr)^2, we see that \Delta_\phi is the Star Graph and G_\phi = Z_2 is the cyclic group of order 2. It is easy to see that the monodromy group of \Phi has the generators

 \begin{matrix} \sigma_0 & = & \bigl( 1 \ 5 \ 3 \bigr) \, \bigl(2 \ 4 \ 6) \bigr) \\[5pt] \sigma_1 & = & \bigl(3 \ 7 \ 4 \bigr) \\[5pt] \sigma_\infty & = & \bigl( 1 \ 3 \ 2 \ 6 \ 4 \ 7 \ 5 \bigr) \end{matrix}

so that G_\phi = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle = A_7 is the alternating group.  Here’s the strange part. The monodromy group of the composition \beta = \phi \circ \Phi has the generators

\begin{matrix} \sigma_0 & = & \bigl( 3 \ 7 \ 5 \bigr) \, \bigl( 4 \ 6 \ 8 \bigr) \, \bigl( 11 \ 13 \ 12 \bigr) \\[5pt] \sigma_1 & = & \bigl( 1 \ 3 \bigr) \, \bigl( 2 \ 4 \bigr) \, \bigl( 5 \ 11 \bigr) \, \bigl( 6 \ 12 \bigr) \, \bigl( 7 \ 9 \bigr) \, \bigl( 8 \ 10 \bigr) \, \bigl( 13 \ 14 \bigr) \\[5pt] \sigma_\infty & = & \bigl( 1 \ 5 \ 12 \ 4 \ 2 \ 8 \ 10 \ 6 \ 13 \ 14 \ 11 \ 7 \ 9 \ 3 \bigr) \end{matrix}

so that G_\beta = \left \langle \sigma_0, \, \sigma_1, \, \sigma_\infty \right \rangle is a group of order |G_\beta| = 12 \, 700 \, 800 = |G_\phi| \cdot |G_\Phi|^2.

Question\beta = \phi \circ \Phi is a composition of Shabat polynomials, where G_\phi = Z_2 and G_\Phi = A_7. Since |G_\beta| = |G_\phi| \cdot |G_\Phi|^2, do we have G_\beta = Z_2 \ltimes \bigl( A_7 \times A_7 \bigr) as the wreath product of G_\Phi by G_\phi?

I’m really confused by this example because it seems to be part of a general phenomenon…  I will post more on this later as I come up with more examples — and possibly a proof!

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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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