All Questions
Tagged with co.combinatorics homological-algebra
25 questions with no upvoted or accepted answers
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,...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k
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_{...
- 26.5k
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 ...
- 26.5k
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 $K_0(A)$, ...
- 26.5k
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 ...
- 26.5k
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 $F(A) \geq gldim(A)$ ...
- 26.5k
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 ...
- 26.5k
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....
- 26.5k
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 ...
- 10.5k
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 ...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k
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 $P$ ...
- 26.5k
3
votes
0
answers
146
views
Acyclic extensions of acyclic simplicial complexes
Say an abstract simplicial complex $X$ is acyclic if its reduced integral simplicial homology groups $\tilde{\mathrm{H}}^{\Delta}_p(X)$ vanish for all $p\geq 0$. Is it the case that, for any $n>0$, ...
- 61
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.
...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k
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 ...
- 26.5k
2
votes
0
answers
45
views
$K_0$-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 ...
- 26.5k
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 ...
- 26.5k
1
vote
0
answers
118
views
An algebra with two multiplications, based on series-parallel diagrams?
Here is a commutative, unital, associative algebra $\mathcal{F}$ with two ways to multiply. The multiplications come from a construction with Boolean operations and series-parallel diagrams. I want ...
- 1,277
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 $ ...
- 26.5k
1
vote
0
answers
80
views
Extending Reedy dimension to augmented chain complexes of abelian groups
Recall that a normal continuously-graded finite interval is given by a pair $a=([a],f)$ consisting of:
1.) A finite totally ordered set $[a]=[a_0< \dots < a_n]$
2.) A grading function $f:U[a] \...
- 11