Questions tagged [algebraic-combinatorics]

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2
votes
0answers
67 views

Geometric or combinatorial interpretations of the (weak) Bruhat order?

$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
10
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1answer
156 views

$2$-adic valuation of Schur $P$-functions in the power-sum basis

For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...
15
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2answers
574 views

The finite groups with a zero entry in each column of its character table (except the first one)

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
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1answer
163 views

What is a toric lattice? [closed]

What is a toric lattice? and how can I construct one in Macaulay2 and compute its basis? is there any alternative method to make one? Since I went through the whole ...
6
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0answers
69 views

Littelmann Path model and RSK e and f operators

The Littelmann path model defines $e_i$ and $f_i$ operators which correspond in type A to the $e_i$ and $f_i$ operators on semistandard Young tableaux (i.e. since they both are different ways of ...
8
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0answers
144 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
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0answers
177 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
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0answers
97 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 ...
6
votes
2answers
415 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 $f$ ...
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0answers
45 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
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1answer
504 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)^{...
8
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0answers
334 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
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1answer
200 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 ...
4
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0answers
134 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,...
5
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0answers
65 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 \...
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0answers
56 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
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2answers
437 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
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1answer
136 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 ...
6
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0answers
171 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 ...
3
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3answers
358 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 long ago and now I got it. 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 ...
5
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1answer
165 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
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0answers
101 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
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0answers
164 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
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0answers
98 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
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2answers
744 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
1answer
610 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
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0answers
61 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
1answer
265 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
2answers
241 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....
4
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0answers
105 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
1answer
330 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 $f$...
13
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0answers
165 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
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0answers
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
1answer
131 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 $...
3
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0answers
71 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
2answers
382 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
2answers
412 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
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0answers
74 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
1answer
568 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
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2answers
345 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 $x_1,x_2,...,x_n$ ...
10
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0answers
138 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
1answer
174 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
1answer
178 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
0answers
158 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
0answers
172 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
1answer
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
2answers
313 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
1answer
193 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
1answer
248 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 ...
2
votes
2answers
232 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 ...

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