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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 ...
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 ...
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/...
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(...
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 ...
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 ...
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 ...
15 votes
2 answers
863 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 $i$...
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 ...
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}^{...
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 ...
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 ...
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 ...
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-...
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\...
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, ...
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 ...
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 ...
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 \...
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 ...