Questions tagged [algebraic-combinatorics]

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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
3 votes
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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
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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
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Shifted schur function and holonomic

Now let us denote by $\Lambda^{*}(n)$ the algebra of polynomials in $x_{1},\ldots,x_{n}$ that become symmetric in new variables $$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n.$$ Here c is a arbitrary fixed ...
GGT's user avatar
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Puzzle in 3D grid with black and white boxes, related to shelling

Consider a $n$ by $n$ by $n$ grid represented by the set of $3$-uples $S=\{1,2,\dots, n\}^3$. A line (resp. slice) of $S$ is a subset of cardinal $n$ (resp. $n^2$) where two components (resp. one ...
Sebastien Palcoux's user avatar
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685 views

Union of the conjugates of maximal subgroups

This post is a generalization of Union of the conjugates of a proper subgroup. Consider an interval $[H,G]$ in the subgroup lattice of the finite group $G$, with $H \neq G$ and such that: (1) $ \...
Sebastien Palcoux's user avatar
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206 views

Hook-content polynomial

Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
GGT's user avatar
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Dimension of the sum of images of transpose

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\im}{\operatorname{im}}$ Given $A,B\in M_{n\times n}(k)$, define $\rank(A,B):=\dim(\im A+\im B)$. I'm looking for results regarding relationships ...
Andy Soffer's user avatar
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386 views

Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
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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 ...
Brendan Mallery's user avatar
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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:...
GGT's user avatar
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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
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Diagonal operator and infinite wedge space formalism

Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it. https://arxiv.org/pdf/math/0207233.pdf ...
GGT's user avatar
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Holonomic modules and Holonomic functions

Let $$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}\prod_{i=1}^{d}G((i-k)h) . $$ I have proved that $ F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}\in \mathbb{C}(h)[[x]]$ is holonomic and arrive at a ...
GGT's user avatar
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Rank-unimodality and Sperner property of higher dimensional partitions

I did a google-search but have not been able to find much reference on this problem. So I am asking it here hoping to get some information. Consider the set of all 4-dimensional Ferrer's diagram ...
Pritam Majumder's user avatar
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An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
teide4's user avatar
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Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$: $$M(z) = \psi(L(z)),~\...
yuting's user avatar
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A combinatoric identity for characters of reductive groups

Let $G$ be a reductive group over an algebraic closed field (of char 0 if necessary). Let $T\subset G$ be a maximal torus and $S=\mathrm{Sym}^*(X(T))$ be the symmetric algebra of characters of $T$. ...
Fyy's user avatar
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References/applications/context for certain polytopes

First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a ...
Igor Makhlin's user avatar
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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:...
Jandré Snyman's user avatar
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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
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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
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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
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1 vote
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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
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An alternative proof of a subgroup lattice characterization of the infinite cyclic group

In Schmidt's book Subgroup lattices of groups, Theorem 1.2.5 states that a group $G$ is cyclic if and only if its subgroup lattice $L(G)$ is distributive and satisfies the maximal condition. Its proof ...
Sebastien Palcoux's user avatar
1 vote
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281 views

On the number of Eulerian orderings

This post is a sequel of Eulerian ordering of the integers modulo n. Let us recall the definition of an Eulerian ordering: Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$....
Sebastien Palcoux's user avatar
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Counting arrangements around a table with constraints

I have $n$ guests seated around a circular table. I want to serve them meals so that given any two guests $u$ and $v$, either i. $u$ and $v$ have different meals, or ii. $u$'s two neighbors have a ...
Stephen Shea's user avatar
1 vote
0 answers
174 views

Holonomic generating function

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define \begin{align} B(d)&= \frac{1}{d!h^{d-1}...
GGT's user avatar
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Hook-content polynomial 2

Recently I have proven the following identity \begin{align} \sum_{\lambda\in \text{different hook of size d}} \frac{1}{d!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-...
GGT's user avatar
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combinatorial ergodicity and promotion

According to J. Propp, T. Roby, and (I believe) others, a cyclic action on a finite set $S$ given by a bijection $\zeta: S \longrightarrow S$ is said to be ${\it ergodic}$ with respect to a statistic ...
A. Leverkuhn's user avatar
1 vote
0 answers
223 views

Does $[H_i , G_i]$ distributive imply $[H_1 \times H_2, G_1 \times G_2]$ modular?

Let $L(G)$ be the subgroup lattice of $G$ and $[H, G]$ an interval in $L(G)$. A lattice $(L, \wedge, \vee)$ is distributive if $a∨(b∧c) = (a∨b) ∧ (a∨c)$, $\forall a,b,c \in L $, and is modular if ...
Sebastien Palcoux's user avatar
1 vote
0 answers
309 views

Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, i....
PLG's user avatar
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Gessel-Viennot theorem

In the paper, page 76, why we need the condition that the subpath lying between lines y=-x and y=k+1 consists entirely of vertical steps?
Mihawk's user avatar
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Relation between $3$-term Plücker relations and more than $3$-term Plücker relations

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...
Jianrong Li's user avatar
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Addition theorem for Schur function in multivariable

Working with the following problem Expansion in Schur function of negative binomial exponent I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur ...
GGT's user avatar
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Algebraic ode of exponential generating series

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n $$ where $$ a_n = \prod_{i=1}^{n}G((i-1)h) $$ We can conclude that the series satisfies a Linear differential ...
GGT's user avatar
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0 votes
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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 ...
Vni Versvs's user avatar
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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
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0 answers
189 views

What breaks down in the theory of affine hyperplane arrangments?

It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...
MadcowD's user avatar
  • 101
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148 views

Methods to get Holonomic functions

Let $a_n$ be a holonomic sequence. By definition, that means there exists a linear differential equation of finite order which annihilates $F(x)$, where $F(x):=\sum a_n x^n$. Similarly let $b_n$, $...
GGT's user avatar
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Abelian centralizer groups (CA-groups)

I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...
Reza Sheikhi 's user avatar

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