jaden reading superhero comics :)
his enthusiasm is contagious ok
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jaden reading superhero comics :)
his enthusiasm is contagious ok
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Exceptional Automorphisms of S_6
The symmetric group S_6 has a special property that S_n does not have for ANY n ≠6. Really? 6, of all numbers?? How odd.
For any group G, and every g in G, the conjugation map f_g: h ↦ g h g^-1 is an automorphism of G. That is, it is an isomorphism from G -> G. The group of these particular automorphisms (under composition) is called the group of inner automorphisms of a group. Did you know that for every symmetric group S_n EXCEPT n=6, this is the entire automorphism group?
For some reason, S_6 has basically ONE (and only one) weird automorphism. What I mean by this is if we denote the inner automorphisms of S_6 by Inn(S_6), then S_6/Inn(S_6) ~= Z_2. (In other words, you can pick a single automorphism sigma such that the automorphisms of S_6 that are not inner automorphisms can be written as a composition of inner automorphisms and this one sigma.)
So we have this weird situation where |Aut(S_2)|=|Aut(C_2)| = 1, which is kinda trivial, |Aut(S_6)| = 2n!, and |Aut(S_n)| = n! otherwise. Kinda weird, huh?
What is also interesting is the proofs that there are no outer automorphisms for n≠6. Basically you can eliminate all but finitely many n, then you can pick off cases, and you are left with an n=6 shaped hole in your cases that CANNOT be filled, which feels so weird to me. If I was proving this myself I'd be going crazy having proven it for every case except 6, and having to resort to some proof which for some reason doesn't work for finitely many values.
Construction
There are so many different ways to construct the weird automorphism of S_6 - I have some links at the bottom. I particularly like the graph theory/geometric ones. Something about using factorisations of K_6 or a dodecahedron just makes the 6 feel more unique. I will admit I don't think I understand what is fundamentally special about 6 enough yet, though, on a philosophical level.
Practically, though, the constructions all basically boil down to the fact you can put a copy of S_5 inside S_6 in a way that isn't the obvious way, which you can only do for n=6. That is the special part.
After that, you have these 6 cosets of S_5 inside S_6 that S_6 acts upon. In other words, each element of S_6 permutes the 6 cosets of S_5 living inside it. But the group of permutations of 6 cosets is S_6. So we have a mapping from elements of S_6 to S_6 - an automorphism! Is this automorphism an inner automorphism? Each construction shows why they are outer differently, but a common theme is to show that the mapping S_6 to S_6 does not take transpositions to transpositions, but inner automorphisms preserve the cycle type.
Some proofs and examples
The wikipedia article. I like the construction about graph partitions. It does not, however, have much detail sadly.
Fortunately, the graph partitions thing from wikipedia is explained here. It's very short and to the point, also quite nice to look at:
Let’s consider the group S6 of permutations of {1,2,3,4,5,6}. For fairly boring reasons, we can find S5 living inside there — the subgroup w
Fairly elementary explanations, followed by more intense ones:
Requires only basic group theory to understand the first few explanations they provide (although it isn't trivial if you just learned group theory). Bonus points for "MyStIc PeNtAgOnS" (capitalisation mine):
We use a simple description of the outer automorphism of S_6 to cleanly describe the invariant theory of six points in P^1, P^2, and P^3.
Allegedly useful, but I can't grab a copy: (I think its on mathscinet under MR1240362 as per David Leep's personal website's publications section)
Combinatorial Structure of the automorphism group of S6 by T.Y. Lam and David B. Leep, Expositiones Mathematicae11 (1993), no. 4, pp. 289-903.
The comment by Matthew Towers here is also interesting:
$$ \begin{array}{|l|c|c|} \hline \text{cycle structure} & \text{number of permutations} & \text{order} \\ \hline 6 & 120 &a
Here’s another example of how an ultraproduct smooths out finite irregularities: of all the finite symmetric groups, only \(\operatorname{Sym}(6)\) has “exotic” (non-inner) automorphisms. If you take an ultraproduct of all the finite symmetric groups, you can examine the ultralimits of automorphisms of the finite symmetric groups, which we call “internal automorphisms” (like internal sets of the hyperreals), and it turns out that every internal automorphism of \[\prod_{\mathcal{U}}\operatorname{Sym}(n)\] is inner.
Furthermore, if CH holds, then by inspecting cardinalities we see there are many external automorphisms (i.e., not writable as an ultralimit of automorphisms) of this ultraproduct, but that even one of these external automorphisms exist is independent of ZFC.
Neat!
Nakajima varieties provide a natural home for geometric representation theory of simply-laced complex simple Lie algebras. Ultimately, Nakajima theory is a theory about the interaction of symplectic geometry and representation theory.
Yiqiang Li
The Lie algebra E₆ may be defined as the algebra of endomorphisms of a 27-dimensional complex vector space MC which annihilate a particular cubic polynomial. This raises a natural question: what is this polynomial? If we choose a basis for MC consisting of weight vectors {Xw } (for some Cartan subalgebra of E₆ ), then any invariant cubic polynomial must be a linear combination of monomials where ∑w + w′ + w″ = 0. The problem is then to determine the coefficients of these monomials. Of course, the problem is not yet well-posed, since we still have a great deal of freedom to scale the basis vectors Xw . If we work over the integers instead of the complex numbers, then much of this freedom disappears. The Z-module M then decomposes as a direct sum of 27 weight spaces which are free Z-modules of rank 1. The generators of these weight spaces are well-defined up to a sign. Using a basis for M consisting of such generators, a little bit of thought shows that the invariant cubic polynomial may be written as a sum where ∑w,w′,w″ = ±1. The problem is now reduced to the determination of the signs w,w′,w″. However, this problem is again ill-posed, since the Xw are only well-defined up to a sign.
Jacob Lurie
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Poincaré Twist by David J. Wright
Colored plot of the argument of the values of an automorphic function associated to a quasifuchsian group.
Spectral Theory of Dynamical Systems
This book introduces some basic topics in the spectral theory of dynamical systems, but also includes advanced topics such as a theorem due to H. Helson and W. Parry, and another due to B. Host. Spectral Theory of Dynamical Systems Moreover, Ornstein's family of mixing rank one automorphisms is described with construction and proof. Systems of imprimitivity, and their relevance to ergodic theory, are discussed. Baire category theorems of ergodic theory, scattered in the literature, are derived in a unified way. Riesz products are considered, and they are used to describe the spectral types and eigenvalues of rank one automorphisms. "Spectral Theory of Dynamical Systems" is the first book devoted exclusively to this subject, moving from introductory material to some topics of current research. The exposition is at a general level and aimed at advanced students and researchers in dynamical systems.