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October 11, 2006

A new theory of mine

Filed under: Uncategorized — vipulnaik @ 2:56 pm

I have this new theory for sequences of objects of various kinds, and I’m trying to figure out what to do with the theory. I haev prepared lots of write-ups on the theory, as well as fanned out my ideas in many directions. But as yet, I haven’t somehow been able to share my idea with others, or bring my write-ups into a cogent and consistent form.

In this blog post, I plan to give a basic outline of the theory, along with references to more detailed write-ups (which I will put on my homepage). Thus begins a rather loose introduction:

Consider the matrix groups GLn(k) where k is a field. For any fixed n, this is the group of invertible matrices of order n. The question I wanted to ask was: what is the relationship between the matrix guops of different orders? There is a nice relationship by block concatenation. Given a matrix in GLm(k) and a matrix in GLn(k) we can obtain a matrix in GLm+n(k) by putting the matrix of order m in the top left corner and the matrix of order n in the bottom right corner, and the remaining entries as zero.

This is a homomorphism GLm(k) X GLn(k) to GL(m+n)(k). If we call this homomorphism Phim,n, then we have some associativity rulse for the mappings Phi, the mappings Phi are all injective, and there aer also some interesting refinement conditions.

This led me to consider the abstract situation: a sequence of groups Gm with m varying over nonnegative integers, along with block concatenation maps Phim,n:Gm X Gn to Gm+n. I assumed conditions of associativity and refinability, and christened the resulting general structure as Addition to Product Sequence (APS). If all the block concatenation maps are injective, then it is termed an Injective Addition to Product Sequence (IAPS).

From the above discussion, the general linear groups over a field (and more generally, over a commutative ring with identity) form an APS.

Question: what are the general properties of APSes? What are the examples of APSes?

A lot of what we do over individual groups can be done over IAPSes of groups. We can defien the cnocept of a sub-IAPS, and a normal sub-IAPS. The quotient of an IAPS by a normal sub-IAPS is again an APS, but the quotient APS may not be injective. The quotient of an IAPS by a sub-IAPS is in general an APS of sets only (not of groups). The quotient is injective if and only if a certain condition called being saturated is satisfied for the sub-IAPS.

Some examples of IAPSes of groups within the matrix algebra setting:

  • The orthogonal groups form a sub-IAPS of the IAPS of general linear groups. That’s because the block concatenation of two orthogonal matrices is an orthogonal matrix. This sub-IAPS is saturated in the following sense: given an orthogonal matix obtained as the block concatenation of two invertible matrices, both the invertible matrices are themselves orthogonal. The quotient space of the general linear IAPS by the orthogonal IAPS forms an IAPS of sets: this can be thought of as the IAPS of smymetric positive definite bilinear forms.
  • The symplectic groups form a sub-IAPS of the IAPS of general linear groups. That’s again because the block concatenation of two symplectic matrices is a symplectic matrix. This is again saturated, and the quotient space is the spcae of nondegenerate alternating forms.
  • The special linear groups form a sub-IAPS of the IAPS of general linear groups. In fact, this is a normal sub-IAPS. The quotient APS is a constant Abelian group with block concatenation simply being the multiplication map within the Abelian group. The sub-IAPS is not saturated, because there can be invertible matrices that are not unimodular, but whose block concatenation is unimodular.
  • Given a homomorphism of rings, there is an induced homomorphism of the corresponding general linear IAPSes. The kernel of this homomorphism is termed an IAPS of congruence subgroups. Here’s the typical example: the ring of integers and the quotient map from that ring of integers to the ring of integers modulo an integer m. The kernel of this quotient map, forms an IAPS, which is called the IAPS of congruence subgroups.

Once I started looking for APSes, I didn’t cease finding them. Roughly the raison d’etre for IAPSes is as follows: take an object and take the sequence of its powers (direct powers or free powers, in some suitable sense). Then, the automorphism groups of these powers form an IAPS of groups. Guess how? Roughly, for the block concatenation map Phim,n, the automorphism of the mth power acts on the first m coordinates and the automorphism of the nth acts on the last n coordinates.

Here are specific situations:

  • The general linear IAPS over a ring R assigns to each n the automorphism group of the free module Rn.
  • The permutation IAPS assigns to each n the symmetric group on n elements. Note that the permutation IAPS can be embedded inside the orthogonal IAPS over any ring.
  • The general affine IAPS over a ring assigns to each n the affine group of order n over the ring, which is the semidirect product of Rn by GL(n,R) under the usual action.
  • The polynomial automorphism IAPS. Fix a base ring (or base field). Then, consider the IAPS whose nth member is the automorphism group of the polynomial ring in n variables over that base ring or field. These form an IAPS. And this IAPS clearly contains the general affine IAPS.
  • The function field automorphism IAPS. Fix a base field. Consider the IAPS whose nth member is the automorphism group of the pure transcendental extension of the base field of transcendence degree n. This IAPS contains the polynomial automorphism IAPS.
  • The free group automorphism IAPS. This is the IAPS that assigns to each n the automorphism group of the free group on n letters.
  • The tensor algebra automorphism IAPS over a base ring or base field. This assigns to each n the automorphism group of the free tensor algebra in n variables over the base ring (or base field).

There are other IAPSes that don’t quite fit into the above framework but arise naturally: for instance, the mapping class groups form IAPSes, the braid groups form IAPSes. And then, various subs of IAPSes can be defined.

What I’m interested in getting out of IAPS theory is the following:

  • See under what circumstances we can come up with suitable notions of determinant, transpose, parabolic structure, unipotent structure and so on.
  • Analyze the conjugacy classes and see under what circumstances we can get a canonical form. For instance, the permutation IAPS has a canonical form for conjugacy classes through the cycle decomposition, while the general linear IAPS over a field has a canonical form for conjugacy classes through the rational canonical form.
  • Try to understand the generating sets and see whether we can get certain special generating sets that are present in members of small index.

Another interesting observation I made is that just like we do representation theory in the general linear IAPS, we can do representation theory of a group in an arbitrary IAPS. COncepts such as direct sum decomposition of representations canbe formulated in the IAPS language. Concepts such as irreducible representation and complete reducibility can be formulated in the language of IAPSes with an additional parabolic structure.

And reversing roles, we can try representing the members of an IAPS inside another IAPS. For instance, we can study the representations of SLn(Fp) in the general linear IAPS over complex numbers. Here, the IAPS theory of both these IAPSes comes out.

By looking at what happens in the case of the representation theory of the permutation IAPS and of the linear IAPS over finite fields, I have tried to see what we can say in general about the representation theory of one IAPS inside the other. I have got some promising frameworks into which the permutation and linear case both fit.

A quick summary:

  • An APS is a sequence of groups indexed by the natural numbers along with block concatenation maps which are homomorphisms from the direct product of two members to the member whose index is the sum of their indices. The block concatenation maps are required to satisfy some conditions, most notably associativity.
  • When the block concatenation maps are injective, I call the APS an injective APS or IAPS.
  • Though I defined an APS of groups, one can also define APS of rings, APS of sets, APS of monoids etc.
  • There are notions of sub-IAPSes and quotient IAPSes. The quotient of an IAPS by a sub-IAPS is again an APS of groups if and only if the sub-IAPS is normal at every member. It is an IAPS if the sub-IAPS is saturated. These are terms I introduced myself.
  • The general linear IAPS has interesting sub-IAPSes: the special linear IAPS (normal but not saturated, the quotient is a constant Abelian group IAPS), the orthogonal IAPS (saturated but not normal), the symplectic IAPS.
  • IAPSes arise as automorphisms of power sequences: the automorphisms of free modules give the general linear IAPS, the automorphisms of free groups give another IAPS, the automorphisms of polynomial rings give the polynomial automorphism IAPS, the automorphisms of the function field give the function field IAPS. Other IAPSes: the braid group, the affine group, the mapping class group.
  • There is a notion of a parabolic structure on a general IAPS, and such a structure comes naturally for IAPSes that arise as automorphisms of power sequences.
  • I am keen on figuring out when additional structure such as determinant, transpose and parabolic structure can be imposed on the IAPS.
  • I am keen on studying representation theory in an arbitrary IAPS, or possibly in an IAPS with a parabolic structure.
  • I am keen on looking at canonical forms for conjugacy classes for arbitrary IAPSes.
  • I am keen on looking at representation theory of the individual members of arbitrary IAPSes, to tie in the similarities between the representation theory of the symmetric groups and of the linear groups over finite fields.

Please do post your comments on the following:

  • Is the general idea of IAPS clear?
  • Does the notion of IAPS seem a useful abstraction (at a conceptual level)?
  • Do IAPSes holda promise of providing uniform tools for studying the very diverse range of IAPSes?
  • Do IAPSes hold a promise of providing a better language for discussing representation thery?
  • What are the aspects that seem to interest you and on which you would like clear expostulation/elaboration?
  • Do you think I should put in the effort of presenting the theory formally or should I wait for something more from it? If so, what kind of thing should I wait for?
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