All Questions
23 questions
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
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
5
votes
1
answer
309
views
On a proof involving Young symmetrizers acting on tensor spaces
I hope this is not too elementary for this site, but I already asked something similar on MSE which has not received any attention whatsoever. I am extremely unfamiliar with the algebraic/...
- 2,792
7
votes
1
answer
326
views
Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$
Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
3
votes
0
answers
56
views
Is the outer automorphism group of a finite poset finite when the Coxeter matrix has finite order?
Let $P$ be a finite connected poset.
The Cartan matrix $C_P$ of $P$ is defined as the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else for $i,j \in P$.
The Coxeter matrix of $P$ is ...
- 26.5k
1
vote
0
answers
131
views
On the order of the Coxeter matrix of a poset
Let $P$ be a finite connected poset.
The Cartan matrix $C_P$ of $P$ is defined as the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else for $i,j \in P$.
The Coxeter matrix of $P$ is ...
- 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
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
12
votes
0
answers
321
views
Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
9
votes
1
answer
384
views
Smith Normal Form of a Cayley Graph of the Symmetric Group
Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of ...
- 525
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 ...
- 26.5k
5
votes
0
answers
357
views
$\text{Determinant}=(\sum \text{Determinant})^2$
Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...
- 43.2k
11
votes
2
answers
558
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
- 26.5k
2
votes
0
answers
126
views
Maximal number of $S_n$-conjugates living in a hyperplane
Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...
- 21
15
votes
2
answers
862
views
What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
- 26.5k
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
6
votes
1
answer
778
views
Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
- 307
4
votes
0
answers
287
views
Eigenvalues of "modified" Johnson scheme via the representation theory of the symmetric group
I am interested in eigenvalues of the following association scheme, which somewhat resembles the Johnson scheme.
Let $n$ and $k\leq n$ be positive integers.
The $n!/(n-k)!$ vertices of the scheme ...
21
votes
0
answers
904
views
Cauchy matrices with elementary symmetric polynomials
$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...
- 28.6k
28
votes
4
answers
2k
views
Matrices: characterizing pairs $(AB, BA)$
Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
- 6,919
6
votes
1
answer
301
views
Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 2,179
8
votes
1
answer
593
views
Representability of polymatroids over $GF(2)$
A polymatroid is a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$,
2) $A \subset B$ implies $d(A) \leq d(B)$, and
3) $d(A \cap B) + d(A \cup B) \leq d(...
- 25.5k