Let denote the area of matrices with integer entries, and let be the group of invertible matrices with integer entries. The Smith regular type takes an arbitrary matrix and factorises it as , the place , , and is an oblong diagonal matrix, by which we imply that the principal minor is diagonal, with all different entries zero. Moreover the diagonal entries of are for some (which can be the rank of ) with the numbers (referred to as the *invariant components*) principal divisors with . The invariant components are uniquely decided; however there could be some freedom to change the invertible matrices . The Smith regular type could be computed simply; for example, in SAGE, it may be computed calling the perform from the matrix class. The Smith regular type can be accessible for different principal splendid domains than the integers, however we’ll solely be centered on the integer case right here. For the needs of this submit, we’ll view the Smith regular type as a primitive operation on matrices that may be invoked as a “black field”.

On this submit I want to file learn how to use the Smith regular type to computationally manipulate two carefully associated courses of objects:

- Subgroups of a regular lattice (or
*lattice subgroups*for brief); - Closed subgroups of a regular torus (or
*closed torus subgroups*for brief).

(This arose for me because of the want to really carry out (with a collaborator) some numerical calculations with various lattice subgroups and closed torus subgroups.) It’s potential that each one of those operations are already encoded in some present object courses in a computational algebra package deal; I’d have an interest to know of such packages and courses for lattice subgroups or closed torus subgroups within the feedback.

The above two courses of objects are isomorphic to one another by Pontryagin duality: if is a lattice subgroup, then the orthogonal complement

is a closed torus subgroup (with the same old Fourier pairing); conversely, if is a closed torus subgroup, then

is a lattice subgroup. These two operations invert one another: and .

Instance 1The orthogonal complement of the lattice subgroupis the closed torus subgroup

and conversely.

Allow us to focus first on lattice subgroups . As all such subgroups are finitely generated abelian teams, one method to describe a lattice subgroup is to specify a set of turbines of . Equivalently, now we have

the place is the matrix whose columns are . Making use of the Smith regular type , we conclude that

so specifically is isomorphic (with respect to the automorphism group of ) to . Specifically, we see that is a free abelian group of rank , the place is the rank of (or ). This illustration additionally permits one to trim the illustration right down to , the place is the matrix fashioned from the left columns of ; the columns of then give a foundation for . Allow us to name this a *trimmed illustration* of .

Instance 2Let be the lattice subgroup generated by , , , thus with . A Smith regular type for is given byso is a rank two lattice with a foundation of and (and the invariant components are and ). The trimmed illustration is

There are different Smith regular varieties for , giving barely totally different representations right here, however the rank and invariant components will at all times be the identical.

By the above dialogue we are able to symbolize a lattice subgroup by a matrix for some ; this illustration just isn’t distinctive, however we’ll tackle this problem shortly. For now, we deal with the query of learn how to use such information representations of subgroups to carry out primary operations on lattice subgroups. There are some operations which can be very simple to carry out utilizing this information illustration:

One may use Smith regular type to detect when one lattice subgroup is a subgroup of one other lattice subgroup . Utilizing Smith regular type factorization , with invariant components , the relation is equal after some manipulation to

The group is generated by the columns of , so this provides a take a look at to find out whether or not : the row of should be divisible by for , and all different rows should vanish.

Instance 3To check whether or not the lattice subgroup generated by and is contained within the lattice subgroup from Instance 2, we write as with , and observe thatThe primary row is in fact divisible by , and the final row vanishes as required, however the second row just isn’t divisible by , so just isn’t contained in (however is); additionally an analogous computation verifies that is conversely contained in .

One can now take a look at whether or not by testing whether or not and concurrently maintain (there could also be extra environment friendly methods to do that, however that is already computationally manageable in lots of functions). This in precept addresses the problem of non-uniqueness of illustration of a subgroup within the type .

Subsequent, we take into account the query of representing the intersection of two subgroups within the type for some and . We are able to write

the place is the matrix fashioned by concatenating and , and equally for (right here we use the change of variable ). We apply the Smith regular type to to put in writing

the place , , with of rank . We are able to then write

(making the change of variables ). Thus we are able to write the place consists of the correct columns of .

Instance 4With the lattice from Instance 2, we will compute the intersection of with the subgroup , which one may write as with . We acquire a Smith regular typeso . We’ve

and so we are able to write the place

One can trim this illustration if desired, for example by deleting the primary column of (and changing with ). Thus the intersection of with is the rank one subgroup generated by .

An analogous calculation permits one to symbolize the pullback of a subgroup through a linear transformation , since

the place is the concatenation of the identification matrix and the zero matrix. Making use of the Smith regular type to put in writing with of rank , the identical argument as earlier than permits us to put in writing the place consists of the correct columns of .

Amongst different issues, this enables one to explain lattices given by techniques of linear equations and congruences within the format. Certainly, the set of lattice vectors that clear up the system of congruences

for , some pure numbers , and a few lattice vectors , along with an extra system of equations

for and a few lattice vectors , could be written as the place is the matrix with rows , and is the diagonal matrix with diagonal entries . Conversely, any subgroup could be described on this type by first utilizing the trimmed illustration , at which level membership of a lattice vector in is seen to be equal to the congruences

for (the place is the rank, are the invariant components, and is the usual foundation of ) along with the equations

for . Thus one can acquire a illustration within the type (1), (2) with , and to be the rows of so as.

Instance 5With the lattice subgroup from Instance 2, now we have , and so consists of these triples which obey the (redundant) congruencethe congruence

and the identification

Conversely, one can use the above process to transform the above system of congruences and identities again right into a type (although relying on which Smith regular type one chooses, the top outcome could also be a distinct illustration of the identical lattice group ).

Now we apply Pontryagin duality. We declare the identification

for any (the place induces a homomorphism from to within the apparent vogue). This may be verified by direct computation when is a (rectangular) diagonal matrix, and the overall case then simply follows from a Smith regular type computation (one can presumably additionally derive it from the category-theoretic properties of Pontryagin duality, though I can’t accomplish that right here). So closed torus subgroups which can be outlined by a system of linear equations (over , with integer coefficients) are represented within the type of an orthogonal complement of a lattice subgroup. Utilizing the trimmed type , we see that

giving an express illustration “in coordinates” of such a closed torus subgroup. Specifically we are able to learn off the isomorphism class of a closed torus subgroup because the product of a finite variety of cyclic teams and a torus:

Instance 6The orthogonal complement of the lattice subgroup from Instance 2 is the closed torus subgrouputilizing the trimmed illustration of , one can simplify this a bit of to

and one may write this because the picture of the group beneath the torus isomorphism

In different phrases, one can write

in order that is isomorphic to .

We are able to now dualize all the earlier computable operations on subgroups of to supply computable operations on closed subgroups of . As an example:

Instance 7Suppose one needs to compute the sum of the closed torus subgroup from Instance 6 with the closed torus subgroup . This latter group is the orthogonal complement of the lattice subgroup thought-about in Instance 4. Thus now we have the place is the matrix from Instance 6; discarding the zero column, we thus have