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edit: I added conjecture 2 that looks much more accessible.

Here is the elementary combinatorial translation of the problem (read below for the homological background):

Let $n \geq 2$. A Nakayama algebra is a list $[c_0,c_1,...,c_{n-1}]$ of $n$ integers with $c_i \geq 2$, $c_{i+1} \geq c_i-1$ and $c_{n-1}=c_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$. A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c_i$. A proper module is a module with $k < c_i$. A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c_i) \leq s+(t-1)n$ (this should be equivalent to $Ext_A^1(M,M) \neq 0$ ). Define the $i$-th syzygy of a proper module $(i,k)$ as follows: $\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$. A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.

The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):

Conjecture: Any weird module is quasi-periodic.

 

Conjecture 2: In case $M$ is weird, also $\Omega^1(M)$ is weird.

Conjecture 2 implies conjecture 1 and looks much easier but I cant find a good argument yet.

Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.

Homological background:

The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.

Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.

I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers). I should note that maybe there is an easy argument and Im too blind to see it at the moment.

edit: I added conjecture 2 that looks much more accessible.

Here is the elementary combinatorial translation of the problem (read below for the homological background):

Let $n \geq 2$. A Nakayama algebra is a list $[c_0,c_1,...,c_{n-1}]$ of $n$ integers with $c_i \geq 2$, $c_{i+1} \geq c_i-1$ and $c_{n-1}=c_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$. A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c_i$. A proper module is a module with $k < c_i$. A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c_i) \leq s+(t-1)n$ (this should be equivalent to $Ext_A^1(M,M) \neq 0$ ). Define the $i$-th syzygy of a proper module $(i,k)$ as follows: $\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$. A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.

The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):

Conjecture: Any weird module is quasi-periodic.

Conjecture 2: In case $M$ is weird, also $\Omega^1(M)$ is weird.

Conjecture 2 implies conjecture 1 and looks much easier but I cant find a good argument yet.

Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.

Homological background:

The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.

Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.

I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers). I should note that maybe there is an easy argument and Im too blind to see it at the moment.

edit: I added conjecture 2 that looks much more accessible.

Here is the elementary combinatorial translation of the problem (read below for the homological background):

Let $n \geq 2$. A Nakayama algebra is a list $[c_0,c_1,...,c_{n-1}]$ of $n$ integers with $c_i \geq 2$, $c_{i+1} \geq c_i+1$ and $c_{n-1}=c_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$. A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c_i$. A proper module is a module with $k < c_i$. A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c_i) \leq s+(t-1)n$ (this should be equivalent to $Ext_A^1(M,M) \neq 0$ ). Define the $i$-th syzygy of a proper module $(i,k)$ as follows: $\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$. A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.

The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):

Conjecture: Any weird module is quasi-periodic.

Conjecture 2: In case $M$ is weird, also $\Omega^1(M)$ is weird.

Conjecture 2 implies conjecture 1 and looks much easier but I cant find a good argument yet.

Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.

Homological background:

The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.

Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.

I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers). I should note that maybe there is an easy argument and Im too blind to see it at the moment.

edit: I added conjecture 2 that looks much more accessible.

Here is the elementary combinatorial translation of the problem (read below for the homological background):

Let $n \geq 2$. A Nakayama algebra is a list $[c_0,c_1,...,c_{n-1}]$ of $n$ integers with $c_i \geq 2$, $c_{i+1} \geq c_i-1$ and $c_{n-1}=c_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$. A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c_i$. A proper module is a module with $k < c_i$. A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c_i) \leq s+(t-1)n$ (this should be equivalent to $Ext_A^1(M,M) \neq 0$ ). Define the $i$-th syzygy of a proper module $(i,k)$ as follows: $\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$. A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.

The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):

Conjecture: Any weird module is quasi-periodic.

Conjecture 2: In case $M$ is weird, also $\Omega^1(M)$ is weird.

Conjecture 2 implies conjecture 1 and looks much easier but I cant find a good argument yet.

Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.

Homological background:

The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.

Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.

I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers). I should note that maybe there is an easy argument and Im too blind to see it at the moment.

Here is the elementary combinatorial translation of the problem (read below for the homological background):

Let $n \geq 2$. A Nakayama algebra is a list $[c_0,c_1,...,c_{n-1}]$ of $n$ integers with $c_i \geq 2$, $c_{i+1} \geq c_i+1$ and $c_{n-1}=c_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$. A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c_i$. A proper module is a module with $k < c_i$. A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c_i) \leq s+(t-1)n$ (this should be equivalent to $Ext_A^1(M,M) \neq 0$ ). Define the $i$-th syzygy of a proper module $(i,k)$ as follows: $\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$. A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.

The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):

Conjecture: Any weird module is quasi-periodic.

Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.

Homological background:

The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.

Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.

I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers). I should note that maybe there is an easy argument and Im too blind to see it at the moment.

edit: I added conjecture 2 that looks much more accessible.

Here is the elementary combinatorial translation of the problem (read below for the homological background):

Let $n \geq 2$. A Nakayama algebra is a list $[c_0,c_1,...,c_{n-1}]$ of $n$ integers with $c_i \geq 2$, $c_{i+1} \geq c_i+1$ and $c_{n-1}=c_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$. A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c_i$. A proper module is a module with $k < c_i$. A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c_i) \leq s+(t-1)n$ (this should be equivalent to $Ext_A^1(M,M) \neq 0$ ). Define the $i$-th syzygy of a proper module $(i,k)$ as follows: $\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$. A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.

The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):

Conjecture: Any weird module is quasi-periodic.

Conjecture 2: In case $M$ is weird, also $\Omega^1(M)$ is weird.

Conjecture 2 implies conjecture 1 and looks much easier but I cant find a good argument yet.

Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.

Homological background:

The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.

Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.

I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers). I should note that maybe there is an easy argument and Im too blind to see it at the moment.