Questions tagged [algebraic-combinatorics]

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9 votes
1 answer
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Is there a 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 ...
Erik Walsberg's user avatar
3 votes
0 answers
119 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 (...
Bishal Deb's user avatar
14 votes
2 answers
836 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^{...
Amritanshu Prasad's user avatar
7 votes
1 answer
734 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-...
Dr. Evil's user avatar
  • 2,641
15 votes
1 answer
447 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 $X\subseteq E$ ...
Jeremy Martin's user avatar
4 votes
2 answers
275 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....
MIQ's user avatar
  • 83
4 votes
0 answers
131 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 ...
ArB's user avatar
  • 688
10 votes
1 answer
456 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 $f$...
MMM's user avatar
  • 245
13 votes
0 answers
187 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 ...
ArB's user avatar
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2 votes
0 answers
71 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 $(0,1,0,1) = (0,1)^2$ ...
Ehud Meir's user avatar
  • 4,969
4 votes
1 answer
186 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 $...
Antoine Labelle's user avatar
3 votes
0 answers
83 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 ...
darij grinberg's user avatar
8 votes
2 answers
460 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 $n$ ...
Sebastien Palcoux's user avatar
13 votes
2 answers
609 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 $\sigma(C_n) = \sigma(n)$, ...
Sebastien Palcoux's user avatar
0 votes
0 answers
78 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 ...
moshe noiman's user avatar
5 votes
2 answers
2k 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 ...
nobody's user avatar
  • 407
9 votes
2 answers
518 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 $x_1,x_2,...,x_n$ ...
Karthik C's user avatar
  • 261
10 votes
0 answers
204 views

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

If one iterates the multiplicity-free branching rule from $\operatorname{GL}(n)$ representations (finite-dim, over $\mathbb C$) to $\operatorname{GL}(n-1)$ all the way down to $\operatorname{GL}(0)$, ...
Allen Knutson's user avatar
4 votes
1 answer
248 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 $\Delta$ (...
equin's user avatar
  • 181
2 votes
1 answer
190 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 ...
jack's user avatar
  • 2,929
10 votes
0 answers
219 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, ...
DamienC's user avatar
  • 8,103
1 vote
0 answers
177 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,\...
GGT's user avatar
  • 685
34 votes
1 answer
3k 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
342 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 ...
Sebastien Palcoux's user avatar
4 votes
1 answer
320 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 ...
Zach H's user avatar
  • 1,899
5 votes
1 answer
300 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 ...
Sebastien Palcoux's user avatar
3 votes
3 answers
422 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 ...
Sebastien Palcoux's user avatar
3 votes
0 answers
124 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? ...
Richard Stanley's user avatar
1 vote
0 answers
88 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 ...
squiggles's user avatar
  • 238
2 votes
1 answer
275 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|...
QuantumMechanic's user avatar
3 votes
2 answers
392 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 ...
Joakim Uhlin's user avatar
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 ...
Sebastien Palcoux's user avatar
14 votes
2 answers
1k views

On the finite simple groups with an irreducible complex representation of a given dimension

This answer of Geoff Robinson shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $3$ if and only if it is isomorphic to $A_5$ or $\mathrm{PSL}(2,7)$. ...
Sebastien Palcoux's user avatar
3 votes
0 answers
108 views

Decomposition of Schur modules over the orthogonal group

Let $V=\mathbb{R}^n$ and $O(n)$ the orthogonal group acting with its standard action on $V$. Now for any partition $\lambda$ we have the Schur module $S_\lambda V$ which is a representation of $O(n)$. ...
Hans's user avatar
  • 2,863
5 votes
1 answer
275 views

Jack polynomial and Selberg integral

I am studying the generalisation of the Selberg's integral by using Jack polynomial as outlined by Forrester and Warnaar (arXiv: 0710.3981). They write down the symmetric Jack polynomial as \begin{...
morgoth's user avatar
  • 53
9 votes
2 answers
793 views

Groups without factorization

A group G is said to have a factorization if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \ | \ a \in A, b \in B \}$. The paper Factorisations of sporadic simple groups (...
Sebastien Palcoux's user avatar
3 votes
1 answer
181 views

Maximal factorization of finite simple groups and no extra intermediate

The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,...
Sebastien Palcoux's user avatar
0 votes
1 answer
246 views

If the coefficient of the polynomial positive

I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$ $$\bar{S}(k)=\...
GGT's user avatar
  • 685
3 votes
1 answer
328 views

Infinite group generated by a single coset

Let $G$ be an infinite countable group having a core-free subgroup $H$ such that the interval $[H,G]$ in the subgroup lattice $\mathcal{L}(G)$ is ACC of infinite length, and for every $K \in (H,G]$, $...
Sebastien Palcoux's user avatar
2 votes
0 answers
66 views

Annihilator of the of the generating function not holonomic

The following is a generating function in $x,h$ with infinite parameters $q_1,q_2\ldots,$ and $w_1, w_2,\ldots$. $$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
GGT's user avatar
  • 685
9 votes
1 answer
356 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 ...
Nathan Lindzey's user avatar
10 votes
1 answer
318 views

About $K$-rectification of increasing tableaux

Let $T$ be a standard Young tableaux on $[n]$. Denote the RSK algorithm $\text{RSK}(w)=(P(T),Q(T))$ for $w\in\mathfrak{S}_n$, where $P(T)$ is the Schencted insertion tableaux. For $1\leq i\leq j\leq n$...
Sylvester W. Zhang's user avatar
1 vote
0 answers
99 views

About parabolic Kazhdan-Lusztig polynomials

Denote by $P^{I,y}_{x,w}$ be the parabolic Kazhdan-Lusztig polynomial of ${}^IW$ of type $y$. I have heard that the polynomials $P^{I,q}_{x,w}$ give the transition matrix between a canonical basis ...
James Cheung's user avatar
  • 1,855
4 votes
1 answer
340 views

Kazhdan–Lusztig polynomials in terms of Ext groups

Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest ...
James Cheung's user avatar
  • 1,855
5 votes
2 answers
728 views

Plucker relations in orthogonal Grassmannian

Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $\varpi_3$. Since $\varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known ...
icmes imrf's user avatar
8 votes
0 answers
150 views

Continuous analogues of Schützenberger promotion

Has anyone studied continuous analogues of Schützenberger promotion, and in particular, a flow on (a suitable subset of) the order polytope of a poset? Here’s what I have in mind: Given a poset $P$, ...
James Propp's user avatar
  • 19.4k
6 votes
1 answer
388 views

hook-length formula: "Fibonaccized": Part II

This is a natural follow-up to my previous MO question, which I share with Brian Hopkins. Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \...
T. Amdeberhan's user avatar
15 votes
2 answers
1k views

hook-length formula: "Fibonaccized" Part I

Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \lambda$, define the hook numbers $h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1$ where $\...
T. Amdeberhan's user avatar
1 vote
0 answers
29 views

Extension of definition of Holonomic closure

My question is about finding the annihilator of a series. Let me begin with what is known and then ask my question. Let $s_d(\frac{q_1}{h},\ldots )$ denote schur function for partition $\lambda =[d]$ ...
GGT's user avatar
  • 685
10 votes
1 answer
2k views

Open problems concerning all the finite groups

What are the open problems concerning all the finite groups? The references will be appreciated. Here are two examples: Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy ...