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Questions: 1. Is there a nice condition/classification when $T=(w,v)$ has finite global dimension for a given $w$? Is there a closed formula for the determinant of $M_T$ in general?

 

2. How many such tuples of finite global dimension exist for a given $n$ up to isomorphism?

3. (edited, special case of 1.) Given n, for which $w<n$ (we can assume $w<n$ since for $w=n$ it always exists) does exist a tuple $(w,v)$ such that $M_T$ has determinant 1?

Questions: 1. Is there a nice condition/classification when $T=(w,v)$ has finite global dimension for a given $w$? Is there a closed formula for the determinant of $M_T$ in general?

2. How many such tuples of finite global dimension exist for a given $n$ up to isomorphism?

3. (edited, special case of 1.) Given n, for which $w<n$ (we can assume $w<n$ since for $w=n$ it always exists) does exist a tuple $(w,v)$ such that $M_T$ has determinant 1?

3. (edited, special case of 1.) Given n, for which $w<n$ (we can assume $w<n$ since for $w=n$ it always exists) does exist a tuple $(w,v)$ such that $M_T$ has determinant 1? Computer experiments suggest for $n \geq 4$ the answer is that such $T$ exists if and only if $w \in \{2,3,..., \frac{n+1}{2} \}$ in case $n$ is odd and $w \in \{2,3,..., \frac{n+2}{2} \}$ in case $n$ is even.

3. (edited, special case of 1.) Given n, for which $w<n$ (we can assume $w<n$ since for $w=n$ it always exists) does exist a tuple $(w,v)$ such that $M_T$ has determinant 1?