37
votes
Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics?
I'm posting a community wiki answer to compile all the known information about the status of all the problems.
Problem 1 (The Generalized Lower Bound Theorem/g-theorem for Gorenstein* complexes): In ...
Community wiki
25
votes
Accepted
A new combinatorial property for the character table of a finite group?
By standard manipulations with the group algebra, your sum has a combinatorial/probabilistic interpretation that makes its nonnegativity clear.
The element $ \frac{1}{|G|} \sum_{ g\in G} [g hg^{-1} ]$...
21
votes
Accepted
On the finite simple groups with an irreducible complex representation of a given dimension
I will write this an an answer, though the answer to the basic question is provided by one of the oldest results in finite group theory.
There is a theorem of C. Jordan, proved in the 19th century, ...
19
votes
How to constructively/combinatorially prove Schur-Weyl duality?
This is a quick answer to explain the statement that the hard direction of Schur-Weyl duality is the same thing the First Fundamental Theorem of invariant theory.
Let $V$ be a finite dimensional ...
19
votes
Have you seen my matroid?
One can use Whitney's theorem to show that the characteristic polynomial is
$$ \sum_{i=0}^k{n\choose i}q^{k-i}(q-2)^{n-i} +
\sum_{i=k+1}^n{n\choose i}(q-2)^{n-i}. $$
I doubt that this can be ...
18
votes
Accepted
Inequality for hook numbers in Young diagrams
Not sure, please check carefully. (Well, now more sure and the argument is more direct.)
I claim that the array $(h)$ majorates the array $(q)$, that is,
$\sum \varphi (h_{ij})\geqslant \sum \varphi(...
17
votes
Accepted
The Möbius number of the nonabelian finite simple groups
The answer to your question is ``yes". Something more general was proved by Hawkes, Isaacs and \"Ozaydin in a 1989 paper in the Rocky Mountain Journal of Mathematics.
CORRECTION: As Sebastien ...
17
votes
Accepted
Groups without factorization
This must be "well-known": If we have $G = AB$ when $G$ is a finite group, and $A,B$ are proper subgroups of $G$, then we may suppose that $A$ and $B$ are both maximal.
For if $A$ is not maximal, and ...
16
votes
Accepted
Is this sum of cycles invertible in $\mathbb QS_n$?
$\newcommand{\cyc}{\operatorname{cyc}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\BB}{\mathbf{B}}
\newcommand{\AA}{\mathbf{A}}
\newcommand{\kk}{\mathbf{k}}
\newcommand{\ww}{\mathbf{w}}
$
PART 1 ...
16
votes
Accepted
The finite groups with a zero entry in each column of its character table (except the first one)
Partial answer: the finite group $G$ is clearly in this class if it has a $p$-block of defect zero for every prime $p$ which divides $|G|$. This is a sufficient condition which may not be necessary. ...
15
votes
Accepted
Identity involving a sum over all partitions of $n$
Here's a quick sketch (since I'm pressed for time). Multiply both sides of the identity by $t^n$ and sum over $n$ from $0$ to infinity. From the cycle decomposition identity (Polya's formula) the ...
15
votes
Accepted
Lang's Jacobian identity: slicker, elementary proof?
Awesome question! I haven't looked at Lang's paper yet, so I can't comment on whether this will be a different approach, but it is elementary. I will make use of Glynn's determinant formula at some ...
15
votes
Accepted
Number of conjugacy classes of finite reductive groups
Assume that $G$ is adjoint split over $\mathbb F_q$. Let $G^*$ be the (Langlands) dual group; it is also split over $\mathbb F_q$. For each semisimple $s \in G^*( \mathbb F_q)$ let $N_s$ be the number ...
14
votes
Why is the catalecticant invariant under coordinate changes?
Dolgachev (2012, p. 57; pdf) observes that your matrix $\left( a_{i+j-2}\right) _{1\leq i\leq n+1,\ 1\leq j\leq n+1}$ (with determinant $\operatorname{Cat} f$) is the matrix of a symmetric bilinear ...
14
votes
Accepted
Have you seen my matroid?
Let $U$ be the uniform matroid of rank $k$ on $n$. Since $U$ is orientable one can consider the Lawrence oriented matroid $\Lambda(U)$ associated with any orientation of $U$ (the Lawrence construction ...
14
votes
How to be rigorous about combinatorial algorithms?
Broadly speaking, there are three approaches to reasoning about software semantics: Denotational semantics provides a mapping from a computer program to a mathematical object representing its meaning. ...
14
votes
On the finite simple groups with an irreducible complex representation of a given dimension
This is just to add some references to Geoff Robinson's answer for classifications of irreducible representations in low dimensions. These are all for quasisimple groups i.e. perfect groups $G$ for ...
13
votes
Accepted
Canon in algebraic combinatorics and how to study
It's a very strange though not unusual idea that one must study the subject before starting to work in it. No, you really don't, at least not in combinatorics.
Let me clarify how the process works. ...
13
votes
Accepted
Abundancy index and non-solvable finite groups
I can answer Questions 1 and 4.
Make sure you look at S. Carnahan's answer. It deals with Questions 2 and 3.
Questions 1 and 4: If a finite group $G$ is not solvable then $|G|$ is divisible by $|G_0|$ ...
12
votes
Plucker relations in orthogonal Grassmannian
If $G = SO(7)$ and $P$ corresponds to $\varpi_3$, then
$$
G/P \cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric)....
12
votes
A new combinatorial property for the character table of a finite group?
This is indeed well-known in the character theory literature, and goes back to Frobenius and Burnside. What you are calculating is a positive rational multiple of a class algebra constant, and class ...
12
votes
Accepted
Do you know an elegant proof for this expression for a Schur function?
I would like to suggest an interpretation using super symmetric functions. These are symmetric functions that are symmetric in two sets of variables $\{x_i\}$ and $\{y_j\}$ separately. They satisfy ...
12
votes
Accepted
What is the centralizer of a Young subgroup of $S_n$?
This answer is largely inspired by the wonderful paper of Samuel Creedon, The Farahat-Higman Algebra of Centralizers of Symmetric Group Algebras, which studies in detail the case of $\mathbb{C}S_n^{S_{...
11
votes
Why is the catalecticant invariant under coordinate changes?
Let $d=2n$ be the degree of your binary form $f$.
Let me introduce $n+1$ pairs of formal variables $\alpha^{(1)}=(\alpha^{(1)}_{1},\alpha^{(1)}_{2}),\ldots, \alpha^{(n+1)}=(\alpha^{(n+1)}_{1},\alpha^{(...
11
votes
Is this sum of cycles invertible in $\mathbb QS_n$?
$\newcommand{\cyc}{\operatorname{cyc}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\BB}{\mathbf{B}}
\newcommand{\AA}{\mathbf{A}}
$
PART 2 OF 3
[This is part of a long answer, which I had to ...
11
votes
Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$
Here is a proof for your identity in Question 3: define the functions
\begin{equation}
F(n,k):=\frac{\left(-1\right)^{k-i} \left(k-1\right)! \left(n-k\right)!}
{2\left(i-1\right)! \left(n-i\right)!} \...
10
votes
Accepted
Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$
It is very probable that what is written below is the simplification of Darij's argument. I use the notation $x^{\underline{n}}=x(x-1)\dots(x-n+1)$ [as in Knuth's books] for the falling factorial, and ...
10
votes
What is the smallest cardinality of a self-linked set in a finite cyclic group?
The difference cover problem has been better studied in the context of $\mathbf{Z}$. Redei, Renyi, and others in the 40s asked for the size of the smallest set $A$ such that $A-A$ covers $\{1,2,\dots,...
10
votes
How few $k$-dimensional subspaces of $V$ are enough to have a complement to each $n-k$-dimensional subspace?
If $F$ is algebraically closed then $d(n,k)=k(n-k)+1$.
For $W\subset V$ of dimension $k$, write $X_W\subset Gr(V,n-k)$ for the set of $n-k$-dimensional subspaces of $V$ that intersect $W$ non-...
10
votes
Accepted
hook-length formula: "Fibonaccized" Part I
Sam is correct of course about $q$-hook formula. Below is a short self-contained proof not relying on such advanced combinatorics.
Denote $h_1>\ldots>h_k$ the set of hook lengths of the first ...
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