# Questions tagged [algebraic-combinatorics]

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234
questions

**7**

votes

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77 views

### Branching Rule for Specht Modules over Kazhdan-Lusztig Basis

Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules
$$S^\...

**5**

votes

**0**answers

140 views

### Can Matsumoto's theorem for the symmetric group be proved using a monovariant?

This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions:
Let $n$ be a nonnegative integer. ...

**0**

votes

**0**answers

89 views

### How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...

**7**

votes

**2**answers

401 views

### Chip-firing clocks

Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ ...

**1**

vote

**0**answers

39 views

### Making the entries of a matrix positive

I am considering two slightly more relaxed version of the question asked here: https://math.stackexchange.com/questions/119034/making-the-entries-of-a-matrix-positive
The two questions are:
Question 1:...

**15**

votes

**0**answers

306 views

### a Vandermonde-type of determinants summed over permutations

Let $S_n$ be the symmetric group. Consider
$$D:=\sum_{\sigma\in S_n} \text{sgn}(\sigma)\cdot \det\begin{pmatrix}1 & a_{\sigma(1)}-0 & (a_{\sigma(1)}-0)^2 & \cdots & (a_{\sigma(1)}-0)^{...

**9**

votes

**0**answers

317 views

### Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...

**4**

votes

**1**answer

195 views

### proof of result from Ian Macdonald's paper “A New Class of Symmetric Functions”

I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...

**3**

votes

**0**answers

122 views

### Does every finite lattice embed into a finite Eulerian lattice?

A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice $L$ such that $\mu(a,b) = (-1)^{|b|-|a|}$ for all $a,...

**4**

votes

**0**answers

57 views

### Form on symmetric functions and their q,t- analogues

[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials]
The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...

**2**

votes

**0**answers

52 views

### Double Schur function expansion

In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion.
\begin{align} \label{eq:...

**9**

votes

**2**answers

433 views

### Characterization of the family of simple groups PSL(2,q) by tensor multiplicity

Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...

**2**

votes

**1**answer

127 views

### Generalised operad structures

We can naively consider an operad as a collection $\{P(n)\}_{n\geq 0}$ of vector spaces $P(n)$ consisting of "functions" with $n$ inputs and one output, equipped with a number of ...

**5**

votes

**0**answers

167 views

### Word/Loop in $L(\Lambda)$

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$).
Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote ...

**2**

votes

**3**answers

274 views

### Given a positive integer $n$, some straight lines and lattice points such… Prove that the number of the lines is at least $n(n+3)$

I was trying to get an answer on MathSE but with no success.
Given a positive integer $n$ and some straight lines in the plane
such that none of the lines passes through $(0,0)$, and such that every ...

**5**

votes

**1**answer

153 views

### Is this Laurent phenomenon explained by invariance/periodicity?

In Chapter 4 of his Tracking the Automatic
Ant, David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana Scott and inspired by the Somos sequences:
Sequence 1. ...

**3**

votes

**0**answers

96 views

### Polynomiality of the inverse of equally weighted Varchenko matrices attached to hyperplane arrangements

Let $d\in \mathbb{Z}_{\ge 1}$, let $\sigma = (H_i)_{i\in \mathcal{I}}$ be a finite hyperplane arrangement in $\mathbb{R}^d$, where $H_i\subset \mathbb{R}^d$ is a hyperplane for $i\in \mathcal{I}$ (the ...

**6**

votes

**0**answers

160 views

### Is there are good algebraic model of random n-hypergraphs?

Suppose $F$ is a finite field and $-1$ is a square in $F$. Let $E$ be the binary relation on $F$ where $(a,b) \in E$ iff $a - b$ is a square. Then $(F,E)$ is called a Paley graph. Paley graphs are ...

**3**

votes

**0**answers

93 views

### Can equality of chromatic symmetric functions of two trees be checked in polynomial time?

Stanley defined chromatic symmetric functions (CSF) in 1995 (Advances in Math) where he conjectured that trees can be distinguished by their CSF. However, tree isomorphism is decidable in P (...

**14**

votes

**2**answers

722 views

### Do you know an elegant proof for this expression for a Schur function?

I know that the identity
$$
s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i}
$$
holds.
Here $\alpha=1^{...

**7**

votes

**1**answer

558 views

### Number of conjugacy classes of finite reductive groups

Let $G$ be a connected reductive group over $\mathbb{Z}$. Let $c_{G(\mathbb{F}_q)}$ be the number of conjugacy classes of $G(\mathbb{F}_q)$.
Question: Is it true that $c_{G(\mathbb{F}_q)}$ is a quasi-...

**0**

votes

**0**answers

59 views

### Is there a cyclic map on unipotent group?

$\DeclareMathOperator{\Gr}{\operatorname{Gr}}\DeclareMathOperator{\SL}{\operatorname{SL}}$In the paper, on page 1, a cyclic map $\rho$ on the Grassmannian $\Gr_{k,n}(\mathbb{C})$ is defined as follows:...

**14**

votes

**1**answer

253 views

### Another characterization of matroids

Has anyone seen the following characterization of matroids?
Let $\Delta$ be a simplicial complex on finite ground set $E$. Then $\Delta$ is a matroid complex if and only if, for every $X\subseteq E$ ...

**4**

votes

**2**answers

236 views

### Combinatorial representation of function

Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0....

**3**

votes

**0**answers

101 views

### A combinatorial proof of an identity of partitions (Macdonald I.5)

This is a statement from Symmetric Functions and Hall Polynomials by Macdonald:
$\sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2$ where $\lambda$ denotes a partition or a Young diagram, and for each ...

**10**

votes

**1**answer

321 views

### Generalization of symmetric functions

A $n$-variable function $f$ is a symmetric function if
$$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$
for every permutation $\sigma \in S_n$.
In particular, if $f$...

**12**

votes

**0**answers

162 views

### Relationship between crystal root operators and usual $e_i, f_i$?

Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...

**2**

votes

**0**answers

69 views

### Bijections between binary sequences and primitive elements in a finite field [duplicate]

Let $n>1$ be a natural number. We call a binary sequence $(b_1,\ldots, b_n)\in \{0,1\}^n$ $rigid$ if it is not a proper power of a sequence of shorter length. So for example $(0,1,0,1) = (0,1)^2$ ...

**4**

votes

**1**answer

123 views

### Digraphs with unique walk of length $k$ between any two vertices

Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices.
Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $...

**4**

votes

**0**answers

70 views

### Subalgebras of a polynomial ring carved out by (families of) coefficient equalities

Let $\mathbf{k}$ be a field, and let $P=\mathbf{k}\left[ x_{1},x_{2}
,\ldots,x_{n}\right] $ be a polynomial ring over $\mathbf{k}$ in $n$
variables $x_{1},x_{2},\ldots,x_{n}$. Alternatively, $P$ can ...

**7**

votes

**2**answers

367 views

### Abundancy index and non-solvable finite groups

Let $\sigma$ be the sum-of-divisors function. A number $n$ is called abundant if $\sigma(n)>2n$. Note that the natural density of the abundant numbers is about $25 \%$. The abundancy index of $n$ ...

**13**

votes

**2**answers

390 views

### On the sum of the subgroup orders of a finite group

Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders
$$\sigma(G) = \sum_{H \le G} |H|.$$
Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...

**0**

votes

**0**answers

71 views

### Minimizing coefficients in a product related to the Rogers Ramanujan identity

Start with the product for partitions into parts congruent to $1$ or $4$ modulo $5$:
$(1 + x + x^2 + x^3 + ...)(1 + x^4 + x^8 + x^{12} +...)(1 + x^6 + x^{12} + x^{18} +...)$...
Now replace some of the ...

**5**

votes

**1**answer

450 views

### Canon in algebraic combinatorics and how to study

1) In subjects such as algebraic geometry, algebraic topology there is a very basic standard canonical syllabus of things one learns in order to get to reading research papers.
Is there a similar ...

**9**

votes

**2**answers

319 views

### Decomposing a polynomial ring into Specht Modules

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...

**9**

votes

**0**answers

126 views

### Branching from $GL(a+b)$ to $GL(a)\times GL(b)$ using Gel'fand-Cetlin patterns

If one iterates the multiplicity-free branching rule from $GL(n)$ representations (finite-dim, over $\mathbb C$) to $GL(n-1)$ all the way down to $GL(0)$, one obtains triangular "Gel'fand-Cetlin (or ...

**4**

votes

**1**answer

160 views

### What Stanley-Reisner rings are $\mathbb{Q}$-Gorenstein?

Let $\Delta$ be a simplicial complex and let $R$ be the associated Stanley-Reisner ring. We can characterize when $R$ is Cohen-Macaulay or when $R$ is Gorenstein in terms of the topology of $\Delta$ (...

**2**

votes

**1**answer

177 views

### Every element of $A$ and $B$ differ in at least $k$ positions

Let $m,n$ be positive integers, $m,n>1$ and $X = \{(x_1,x_2, ..., x_m) \in \mathbb{Z}^m :1 \le x_i \le n, \forall 1 \le i \le m\}$.
$A$ and $B$ are two disjoint subsets of $X$, such that if $a \in ...

**10**

votes

**0**answers

141 views

### What is the meaning of the coefficients of the Alekseev-Torossian associator

Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ...

**1**

vote

**0**answers

171 views

### Combinatorial bijection on monotone sequences

Let $(n),\mu$ be the partition of $n$ define $H_g^{m}((n);\mu)$ count's the number of tuples $(\tau_1,\ldots,\tau_r)$ of transposition in symmetric group $S_n$ with the following conditions
$$ (1,2,\...

**34**

votes

**1**answer

2k views

### Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics?

[I am a co-moderator of the recently started Open Problems in Algebraic Combinatorics blog and as a result starting doing some searching for existing surveys of open problems in algebraic ...

**8**

votes

**2**answers

305 views

### Lower bound for the order of a simple group with a given class number

Every simple group below are assumed non-abelian.
Let us call the class number $k(G)$ of a finite group $G$ the number of its conjugacy classes (also, the number of its irreducible complex ...

**4**

votes

**1**answer

176 views

### Transition equations for double Schubert polynomials

For $w$ a permutation, the associated (ordinary) Schubert polynomial $\mathfrak{S}_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X_w$ in ...

**5**

votes

**1**answer

245 views

### What is the smallest rank for a noncommutative fusion ring?

A fusion ring $\mathcal{F}$ (of rank $r$) is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying ...

**1**

vote

**2**answers

226 views

### Is there a noncommutative simple fusion ring?

A fusion ring $\mathcal{F}$ is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying axioms slightly ...

**3**

votes

**0**answers

116 views

### Number of adjoint orbits containing a $(0,1)$-matrix

Motivated by this
question, what can be said about the number $f(n)$ of adjoint
orbits of $\mathrm{Mat}(n,\mathbb{C})$ (the ring of all $n\times n$
complex matrices) that contain a $(0,1)$-matrix? ...

**1**

vote

**0**answers

74 views

### Flag $f$-vectors of CW-complexes

Hidden away in the appendix of this nice paper by Björner and Kalai, they give a clean description of $f$-vectors that can arise from regular CW-complexes in terms of truncations of the Euler-Poincaré ...

**1**

vote

**1**answer

180 views

### Finding Littlewood-Richardson coefficients without using identities

The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq |Q| + |P|...

**3**

votes

**2**answers

178 views

### NE-Lattice paths from $(0,0)$ to $(n,n)$ with $k$ peaks

Let $n$ and $k$ be natural numbers. I will consider North-East lattice paths (NE-paths) from $(0,0)$ to $(n,n)$ and encode these as strings of length $2n$ with letters $\mathsf{N}$ and $\mathsf{E}$. A ...

**21**

votes

**2**answers

1k views

### A new combinatorial property for the character table of a finite group?

Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character.
Consider the following combinatorial property of $\Lambda$: for ...