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Iguanodon series of simple groups

Published November 7, 2007 by lievenlb

Bruce Westbury has a page on recent work on series of Lie groups including exceptional groups. Moreover, he did put his slides of a recent talk (probably at MPI) online.

Probably, someone considered a similar problem for simple groups. Are there natural constructions leading to a series of finite simple groups including some sporadic groups as special members ? In particular, does the following sequence appear somewhere ?

$L_2(7), M_{12}, A_{16}, M_{24}, A_{28}, A_{40}, A_{48}, A_{60}, \ldots $

Here, $L_2(7) $ is the simple group of order 168 (the automorphism group of the Klein quartic), $M_{12} $ and $M_{24} $ are the sporadic Mathieu groups and the $A_n $ are the alternating simple groups.

I’ve stumbled upon this series playing around with Farey sequences and their associated ‘dessins d’enfants’ (I’ll come back to the details of the construction another time) and have dubbed this sequence the Iguanodon series because the shape of the doodle leading to its first few terms

reminded me of the Iguanodons of Bernissart (btw. this sketch outlines the construction to the experts). Conjecturally, all groups appearing in this sequence are simple and probably all of them (except for the first few) will be alternating.

I did verify that none of the known low-dimensional permutation representations of other sporadic groups appear in the series. However, there are plenty of similar sequences one can construct from the Farey sequences, and it would be nice if one of them would contain the Conway group $Co_1 $. (to be continued)

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M-geometry (3)

Published September 18, 2007 by lievenlb

For any finite dimensional A-representation S we defined before a character $\chi(S) $ which is an linear functional on the noncommutative functions $\mathfrak{g}_A = A/[A,A]_{vect} $ and defined via

$\chi_a(S) = Tr(a | S) $ for all $a \in A $

We would like to have enough such characters to separate simples, that is we would like to have an embedding

$\mathbf{simp}~A \hookrightarrow \mathfrak{g}_A^* $

from the set of all finite dimensional simple A-representations $\mathbf{simp}~A $ into the linear dual of $\mathfrak{g}_A^* $. This is a consequence of the celebrated Artin-Procesi theorem.

Michael Artin was the first person to approach representation theory via algebraic geometry and geometric invariant theory. In his 1969 classical paper “On Azumaya algebras and finite dimensional representations of rings” he introduced the affine scheme $\mathbf{rep}_n~A $ of all n-dimensional representations of A on which the group $GL_n $ acts via basechange, the orbits of which are exactly the isomorphism classes of representations. He went on to use the Hilbert criterium in invariant theory to prove that the closed orbits for this action are exactly the isomorphism classes of semi-simple -dimensional representations. Invariant theory tells us that there are enough invariant polynomials to separate closed orbits, so we would be done if the caracters would generate the ring of invariant polynmials, a statement first conjectured in this paper.

Claudio Procesi was able to prove this conjecture in his 1976 paper “The invariant theory of $n \times n $ matrices” in which he reformulated the fundamental theorems on $GL_n $-invariants to show that the ring of invariant polynomials of m $n \times n $ matrices under simultaneous conjugation is generated by traces of words in the matrices (and even managed to limit the number of letters in the words required to $n^2+1 $). Using the properties of the Reynolds operator in invariant theory it then follows that the same applies to the $GL_n $-action on the representation schemes $\mathbf{rep}_n~A $.

So, let us reformulate their result a bit. Assume the affine $\mathbb{C} $-algebra A is generated by the elements $a_1,\ldots,a_m $ then we define a necklace to be an equivalence class of words in the $a_i $, where two words are equivalent iff they are the same upto cyclic permutation of letters. For example $a_1a_2^2a_1a_3 $ and $a_2a_1a_3a_1a_2 $ determine the same necklace. Remark that traces of different words corresponding to the same necklace have the same value and that the noncommutative functions $\mathfrak{g}_A $ are spanned by necklaces.

The Artin-Procesi theorem then asserts that if S and T are non-isomorphic simple A-representations, then $\chi(S) \not= \chi(T) $ as elements of $\mathfrak{g}_A^* $ and even that they differ on a necklace in the generators of A of length at most $n^2+1 $. Phrased differently, the array of characters of simples evaluated at necklaces is a substitute for the clasical character-table in finite group theory.

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M-geometry (2)

Published September 17, 2007 by lievenlb

Last time we introduced the tangent quiver $\vec{t}~A $ of an affine algebra A to be a quiver on the isoclasses of simple finite dimensional representations. When $A=\mathbb{C}[X] $ is the coordinate ring of an affine variety, these vertices are just the points of the variety $X $ and this set has the extra structure of being endowed with the Zariski topology. For a general, possibly noncommutative algebra, we would like to equip the vertices of $\vec{t}~A $ also with a topology.

In the commutative case, the Zariski topology has as its closed sets the common zeroes of a set of polynomials on $X $, so we need to generalize the notion of ‘functions’ the the noncommutative world. The NC-mantra states that we should view the algebra A as the ring of functions on a (usually virtual) noncommutative space. And, face it, for a commutative variety $X $ the algebra $A=\mathbb{C}[X] $ does indeed do the job. Still, this is a red herring.

Let’s consider the easiest noncommutative case, that of the group algebra $\mathbb{C} G $ of a finite group $G $. In this case, the vertices of the tangent quiver $\vec{t}~A $ are the irreducible representations of $G $ and no sane person would consider the full group algebra to be the algebra of functions on this set. However, we do have a good alternative in this case : characters which allow us to separate the irreducibles and are a lot more manageable than the full group algebra. For example, if $G $ is the monster group then the group algebra has dimension approx $8 \times 10^{53} $ whereas there are just 194 characters to consider…

But, can we extend characters to arbitrary noncommutative algebras? and, more important, are there enough of these to separate the simple representations? The first question is easy enough to answer, after all characters are just traces so we can define for every element $a \in A $ and any finite dimensional simple A-representation $S $ the character

$\chi_a(S) = Tr(a | S) $

where $a | S $ is the matrix describing the action of a on S. But, you might say, characters are then just linear functionals on the algebra A so it is natural to view A as the function algebra, right? Wrong! Traces have the nice property that $Tr(ab)=Tr(ba) $ and so they vanish on all commutators $[a,b]=ab-ba $ of A, so characters only carry information of the quotient space

$\mathfrak{g}_A = \frac{A}{[A,A]_{vect}} $

where $[A,A]_{vect} $ is the vectorspace spanned by all commutators (and not the ideal…). If one is too focussed on commutative geometry one misses this essential simplification as clearly for $A=\mathbb{C}[X] $ being a commutative algebra,

$[\mathbb{C}[X],\mathbb{C}[X]]_{vect}=0 $ and therefore in this case $\mathfrak{g}_{\mathbb{C}[X]} = \mathbb{C}[X] $

Ok, but are there enough characters (that is, linear functionals on $\mathfrak{g}_A $, that is elements of the dual space $\mathfrak{g}_A^* $) to separate the simple representations? And, why do I (ab)use Lie-algebra notation $\mathfrak{g}_A $ to denote the vectorspace $A/[A,A]_{vect} $???

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