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M-geometry (3)

By lieven

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

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

from the set of all finite dimensional simple A-representations \wis{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 \wis{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 \wis{rep}_n~A.

So, let us reformulate their result a bit. Assume the affine \C-algebra A is generated by the elements a_1,\hdots,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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3 comments
Posted in geometry
Written on Tue, 18 September 2007 at 8:56 am
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3 Responses to “M-geometry (3)”

  1. Jonathan Vos Post Says:
    October 10th, 2007 at 8:27 am

    ” we would be done if the caracters would” inverse typo transforms to: ” we would be done if the characters would”

  2. Zelah Says:
    October 12th, 2007 at 12:17 pm

    Hi Mr Lieven

    I was wondering if M Geometry had anything to do with Hitchens’s Generalized Geometry aka String theory / M Theory?

  3. lieven Says:
    October 12th, 2007 at 2:13 pm

    Zelah and Javier, here’s how i explained the M in lens a few weeks ago : noncommutative geometry was invented by ringtheorists in the early 70ties, then the terminology was hijacked in the mid 80ties by another group and today everyone claims to be doing it, so it has become empty terminology. ~M \not= NC and that’s about it… in fact, ‘officially’ the M stands for Matrix because we are only looking at finite dimensional representations. unofficially, sometimes Id like the M to stand for My, most of the time though M stands for Minority… Anyway, it seemed important at the time to make the distinction but now that classes have started there are far more things that frustrate me, so i dont care about terminology at the moment.

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