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$K_0$-basis modules with a unique extension related to parking functions

Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points. A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and ...
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3 votes
0 answers
85 views

Exterior powers of the Cartan matrix and Dyck paths

(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
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3 votes
0 answers
112 views

Finite global dimension via the Cartan determinant

Let $A=T(KQ)$ be the trivial extension algebra of a path algebra of Dynkin type $KQ$. The indecomposable module of $A$ correspond to the roots of $Q$ (and not just the positive roots as for $KQ$). Let ...
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9 votes
1 answer
395 views

Which finite posets are Koszul self-dual?

Let $P$ be a finite connected poset with incidence algebra $A_P$. For the definition and results on Koszul algebras for incidence algebras, see for example here Question: Which posets have the ...
Mare's user avatar
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3 votes
0 answers
102 views

Frobenius algebras associated to posets and coalgebra structures

Let $P$ be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m). Let k be a field, then the classical incidence algebra $kP$ has $k$-vector ...
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4 votes
0 answers
83 views

Number of K-generators of an algebra and type $D_n$-parking functions

Let $A$ be a representation-finite quiver algebra. When $A$ has $n$ simple modules a basic module $M$ with $n$ indecomposable summands $M_i$ is called a K-generator when the $M_i$ generate $K_0(A)$, ...
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10 votes
1 answer
396 views

Generalising the union-closed sets conjecture from lattice to a larger class of posets

(edit: I decided to simplify the question and only pose it for bounded posets first) The Union-closed sets conjecture is equivalent for lattices P to: There exists a join-irreducible element $a$ with ...
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3 votes
0 answers
99 views

The union-closed sets conjecture for finite dimensional algebras

Say a finite dimensional algebra $A$ satisfies the right UC-condition if there exists an indecomposable projective module $P$ of $A$ such that $\operatorname{injdim}(\operatorname{top}(P))=1$ and $P$ ...
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7 votes
0 answers
355 views

A homological algebra approach to the Union-closed sets conjecture

I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
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8 votes
0 answers
334 views

Dyck paths of Dynkin type

(The conjecture is a homological algebra question, but question 2 is a pure combinatorics question given that the conjecture is true) A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...
Mare's user avatar
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5 votes
1 answer
226 views

Frobenius algebras from symmetric polynomials

Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...
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4 votes
0 answers
155 views

Commutative algebras associated to simple Lie algebras

In Section 2 of the article https://www.sciencedirect.com/science/article/pii/S0021869307000385, the authors study the center $Z=Z_Q$ of certain preprojective like algebras associated to the simply ...
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5 votes
0 answers
97 views

Periodics of Coxeter matrices for truncated Nakayama algebras

For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$). So for example for $n=7$ and $r=3$ we ...
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10 votes
1 answer
3k views

An enumeration problem for Dyck paths from homological algebra

In their article "On n-Gorenstein rings and Auslander rings of low injective dimension" Fuller and Iwanaga gave a homological characterisation of 2-Gorenstein Nakayama algebras with global ...
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3 votes
0 answers
54 views

Properties of sequences associated to Nakayama algebras

Assume Nakayama algebras are connected and given by quiver and relations. Note that such Nakayama algebras have global dimension at most $2n-2$ in case it is finite and the algebra has $n$ simples. ...
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4 votes
0 answers
73 views

Frobenius dimensions of Nakayama algebras

The Frobenius dimension $F(A)$ of an Artin algebra $A$ is given by the dimension of $Hom(D(A),A)$ (see the answer in On nearly Frobenius algebras ). Question 1: Is it true that $F(A) \geq gldim(A)$ ...
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5 votes
1 answer
268 views

Global dimension 2 Nakayama algebras , 321-avoiding permutations and Fibonacci combinatorics

A $n$-LNakayama algebra (for $n \geq 2$) is a list $[c_0,c_1,...,c_{n-1}]$ with $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i -1 \leq c_{i+1}$ for all $i$. They are in bijection with Dyck paths, ...
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14 votes
2 answers
514 views

Classification of shod Dyck paths

A sequence $[c_0,c_1,...,c_{n-1}]$ with $n \geq 2$ is called a Dyck path in case $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i-1 \leq c_{i+1}$ for each $i$. For example the Dyck paths for $n=4$ ...
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2 votes
1 answer
169 views

Combinatorial problem on periodic dyck paths from homological algebra

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 ...
Mare's user avatar
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7 votes
1 answer
462 views

On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
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11 votes
2 answers
558 views

Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background). A Morita-...
Mare's user avatar
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13 votes
1 answer
745 views

Combinatorial inequality for dominant dimension

In the following I present a conjecture on Nakayama algebras that I have for nearly 2 years now. Since I was not able to solve it and it can be stated purely combinatorically, I thought it might be ...
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8 votes
0 answers
140 views

$n$-fold tensor products of $D(A)$ for finite dimensional algebras

Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected). Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes ...
Mare's user avatar
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4 votes
0 answers
90 views

Number of hereditary modules of a hereditary algebra

Let $Q$ always denote a Dynkin quiver. Given a connected path algebra $A=kQ$ and a module $M$, is there a useful criterion on $M$ when $End_A(M)$ is again a connected quiver algebra? Call a module ...
Mare's user avatar
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4 votes
0 answers
81 views

Sum of all projective dimensions of simple modules

Let $X_{n,t}$ be the set of all finite dimensional algebras (we can assume they are given by a connected quiver and admissible relations) that have global dimension equal to $n$ and $t$ simple modules....
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10 votes
1 answer
400 views

Derived equivalences of Dyck paths

Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top ...
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4 votes
0 answers
210 views

Conjecture on tilting modules for an Auslander algebra

On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism classes of modules, occurring as the $i$-th summand of ...
Tom Copeland's user avatar
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15 votes
2 answers
866 views

What are the periodic Dyck paths?

I changed the thread completely so that everything is now elementary linear algebra. A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
Mare's user avatar
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1 vote
0 answers
77 views

On $Ext_A^2(S,A)$ for algebras $A$

For Nakayama algebras $A$ with a linear quiver and at most 7 points (which are 196 algebras) the following was true: $max \{ dim(Ext_A^2(S,A)) | S $ simple $\}= max \{ dim(Ext_A^2(X,A)) | X $ ...
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8 votes
1 answer
217 views

Categorification of monotone maps via tilting modules?

It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama ...
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3 votes
0 answers
175 views

Geometric interpretation of homological quantities in Artinian local Gorenstein algebras

By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
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1 vote
1 answer
140 views

Is the Cartan permanent odd for finite global dimension?

Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix. Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd ...
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2 votes
0 answers
121 views

Ext of a Schur algebra

Let $A=A_n$ be the representation-finite block of a Schur algebra with $n$ simple modules for $n \geq 2$. Quiver and relations of $A$ can be found in 6.1. of https://arxiv.org/pdf/1607.05965.pdf . Let ...
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3 votes
0 answers
81 views

Number of generalised tilting modules

This question is a modification of Number of tilting modules and was suggested by a comment of Dag Madsen. I wondered about this question too some time ago, but did not do much with it since my ...
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3 votes
0 answers
60 views

Number of syzygy closed Nakayama algebras

Let $A$ be the algebra of $n \times n$ upper triangular matrices over a field $K$ and $J$ its Jacobson radical. Call an ideal $I$ of $A$ admissible in case $I \subseteq J^2$. Let $X$ be the set of ...
Mare's user avatar
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7 votes
0 answers
266 views

Closed formula for some dimension

This question has a background from representation theory/homological algebra, but I state everything in elementary terms here: Call an n-CNak an n-tupel $[c_0,c_1,...,c_{n-1}]$ where $c_0=c_1=...=c_{...
Mare's user avatar
  • 26.5k
14 votes
1 answer
835 views

Special configurations on a circle from a homological algebra problem

Here is the short version of the combinatorial problem: Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
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