Return to Answer

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C_2$, explicitly finding/computing its decomposition into these three representations.

In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page http://www.math.ubc.ca/~cass; he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at http://www.math.ubc.ca/~cass/research/publications, including, for example, The computation of structure constants according to Jacques Tits—things that we all know can be done but that most of us (at least I!) would shrink from actually doing, here laid out in a way that demonstrates how to carry it out practically.

(There's also some nice stuff on mathematical graphics!)

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C_2$, explicitly finding/computing its decomposition into these three representations.

In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page http://www.math.ubc.ca/~cass; he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at http://www.math.ubc.ca/~cass/research/publications.html, including, for example, The computation of structure constants according to Jacques Tits—things that we all know can be done but that most of us (at least I!) would shrink from actually doing, here laid out in a way that demonstrates how to carry it out practically.

(There's also some nice stuff on mathematical graphics!)

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C_2$, explicitly finding/computing its decomposition into these three representations.

In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page http://www.math.ubc.ca/~cass; he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at http://www.math.ubc.ca/~cass/research/publications, including, for example, The computation of structure constants according to Jacques Tits—things that we all know can be done but that most of us (at least I!) would shrink from actually doing, here laid out in a way that demonstrates how to carry it out practically.

(There's also some nice stuff on mathematical graphics!)

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C_2$, explicitly finding/computing its decomposition into these three representations.

In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page http://www.math.ubc.ca/~cass; he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at http://www.math.ubc.ca/~cass/research/publications, including, for example, The computation of structure constants according to Jacques Tits—things that we all know can be done but that most of us (at least I!) would shrink from actually doing, here laid out in a way that demonstrates how to carry it out practically.

(There's also some nice stuff on mathematical graphics!)

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C_2$, explicitly finding/computing its decomposition into these three representations.

In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page http://www.math.ubc.ca/~cass; he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at http://www.math.ubc.ca/~cass/research, including, for example, The computation of structure constants according to Jacques Tits—things that we all know can be done but that most of us (at least I!) would shrink from actually doing, here laid out in a way that demonstrates how to carry it out practically.

(There's also some nice stuff on mathematical graphics!)

In Computing with real tori, Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C_2$, explicitly finding/computing its decomposition into these three representations.

In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page http://www.math.ubc.ca/~cass; he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at http://www.math.ubc.ca/~cass/research/publications, including, for example, The computation of structure constants according to Jacques Tits—things that we all know can be done but that most of us (at least I!) would shrink from actually doing, here laid out in a way that demonstrates how to carry it out practically.

(There's also some nice stuff on mathematical graphics!)