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Najdorf
Najdorf
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A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard notion of irreducibility among GCMs and the common term for it is "indecomposable". Now if you look at most books there is a standard basic classification of GCMs: An indecomposable GCM is of 3 kinds: finite, affine and another category which can be described as all-other-GCMs-that-I-know-little-about-so-I-will-bundle-them-together. There is also another classification based on whether the GCM is "symmetrizable" or not (the finite and affine GCMs are symmetric and hencesymmetric and hence symmetrizable).

What I am looking for is a smart classification of the GCMs that are not finite or affine. For example here is thea number of indefinite GCMs (not finitequestions:

    [1] Is the number of indefinite GCMs (not finite, not affine but symmetrizable) finite? As far as I know indefinite GCMs is a larger set than hyperbolic GCMs which are only finitely many.
    [2] Are all indefinite GCMs invertible?
    [3] If $A$ is indefinite and of size $N$ what is the biggest matrix of finite type that can appear as a principal minor in $A$?

EDIT: corrected a mathematical mistake pointed out by Jim Humphreys, not affine but symmetrizable) finite? As far as I know indefinite is larger set than hyperbolic which are only finitely many. Are they all invertible? If $A$ is indefinite and of size $N$ what isalso improved the biggest matrix of finite type that can appearformat as a principal minor in $A$?to make things more visible. Again suggested by Jim Humphreys.

A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard notion of irreducibility among GCMs and the common term for it is "indecomposable". Now if you look at most books there is a standard basic classification of GCMs: An indecomposable GCM is of 3 kinds: finite, affine and another category which can be described as all-other-GCMs-that-I-know-little-about-so-I-will-bundle-them-together. There is also another classification based on whether the GCM is "symmetrizable" or not (the finite and affine GCMs are symmetric and hence symmetrizable).

What I am looking for is a smart classification of the GCMs that are not finite or affine. For example is the number of indefinite GCMs (not finite, not affine but symmetrizable) finite? As far as I know indefinite is larger set than hyperbolic which are only finitely many. Are they all invertible? If $A$ is indefinite and of size $N$ what is the biggest matrix of finite type that can appear as a principal minor in $A$?

A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard notion of irreducibility among GCMs and the common term for it is "indecomposable". Now if you look at most books there is a standard basic classification of GCMs: An indecomposable GCM is of 3 kinds: finite, affine and another category which can be described as all-other-GCMs-that-I-know-little-about-so-I-will-bundle-them-together. There is also another classification based on whether the GCM is "symmetrizable" or not (the finite and affine GCMs are symmetric and hence symmetrizable).

What I am looking for is a smart classification of the GCMs that are not finite or affine. For example here is a number of questions:

    [1] Is the number of indefinite GCMs (not finite, not affine but symmetrizable) finite? As far as I know indefinite GCMs is a larger set than hyperbolic GCMs which are only finitely many.
    [2] Are all indefinite GCMs invertible?
    [3] If $A$ is indefinite and of size $N$ what is the biggest matrix of finite type that can appear as a principal minor in $A$?

EDIT: corrected a mathematical mistake pointed out by Jim Humphreys, also improved the format as to make things more visible. Again suggested by Jim Humphreys.

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Najdorf
Najdorf
  • 751
  • 7
  • 19

Classification of generalized Cartan matrices (GCMs)

A GCM is square matrix $A = (a_{ij})$ satisfying: (1) $a_{ij} \in \mathbf{Z}$ (2) $a_{ii} = 2$ for all $i$. (3) $a_{ij} \leq 0$ for $i \neq j$. (4) $a_{ij} = 0$ iff $a_{ji} = 0$. There is a standard notion of irreducibility among GCMs and the common term for it is "indecomposable". Now if you look at most books there is a standard basic classification of GCMs: An indecomposable GCM is of 3 kinds: finite, affine and another category which can be described as all-other-GCMs-that-I-know-little-about-so-I-will-bundle-them-together. There is also another classification based on whether the GCM is "symmetrizable" or not (the finite and affine GCMs are symmetric and hence symmetrizable).

What I am looking for is a smart classification of the GCMs that are not finite or affine. For example is the number of indefinite GCMs (not finite, not affine but symmetrizable) finite? As far as I know indefinite is larger set than hyperbolic which are only finitely many. Are they all invertible? If $A$ is indefinite and of size $N$ what is the biggest matrix of finite type that can appear as a principal minor in $A$?