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Questions tagged [algebraic-combinatorics]

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15 views

Hypergraph mapping's projection

I have been struggling quite a while with a question, which I suspect might have a simple answer to: I have a Graph G = (X,E,Ψ) with E (hyperedge) being a family of subsets of X and Ψ being a mapping ...
3
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0answers
92 views

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 ...
2
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1answer
81 views

Frobenius coordinate expansion of character

Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given $$ (a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$ where $d(\lambda)$ denote the diagonal of $\...
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0answers
46 views

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 ...
38
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1answer
766 views

How to be rigorous about combinatorial algorithms?

1. The question This may be the worst question I've ever posed on MathOverflow: broad, open-ended and likely to produce heat. Yet, I think any progress that will be made here will be extremely useful ...
16
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1answer
574 views

Lang's Jacobian identity: slicker, elementary proof?

In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven: Theorem 1. Let $p$ be a prime. Let $\...
16
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2answers
404 views

Eigenvalues and eigenvectors of the matrix with entries $\dbinom{n+1}{2j-i}$ for $i, j = 1, 2, \ldots, n$

Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, ...
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3answers
394 views

Is this sum of cycles invertible in $\mathbb QS_n$?

I am interested the following element of the group algebra $\mathbb{Q}S_n$: \begin{align} \phi_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n) \end{align} where $e$ is the identity permutation. My question ...
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0answers
60 views

Rationality of power series whose coefficients are the ranks of a sequence of matrices

Recently, I stumbled several times about the problem to decide whether a certain formal power series $$ f = \sum_{n=0}^\infty d_n T^n \in \mathbb{Q}[\![T]\!]$$ is actually a rational function, where ...
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0answers
677 views

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 ...
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1answer
203 views

Bijection between noncrossing matchings on $2b$ points and Standard Young Tableaux of size $2 \times b$

I'm currently reading a review article called Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance by Jessica Striker. In this article, Striker writes that there is a ''nice'' ...
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0answers
128 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 ...
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1answer
239 views

Hilbert series of graded Cohen-Macaulay domains, 28 years later?

I am reading through Richard Stanley's 1990 paper "On the Hilbert Function of a Graded Cohen-Macaulay Domain" to present in a seminar. I am trying to provide a reasonable conclusion for this talk, and ...
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0answers
225 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 $C_n := \{0,1, \dots , n-1\}$....
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3answers
709 views

Is the Ford-Fulkerson algorithm a tropical rational function?

The Ford-Fulkerson algorithm Let me recall the standard scenario of flow optimization (for integer flows at least): Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
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0answers
35 views

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 ...
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1answer
535 views

Eulerian ordering of the integers modulo n

Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$. An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that: $$\forall i \le n \ \forall j&...
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2answers
236 views

Is an Eulerian lattice shellable?

The notion of Eulerian lattice generalizes the notion of face lattice of a convex polytope. (Bruggesse-Mani): The boundary complex of a convex polytope is shellable. (Björner-Wachs): A poset is ...
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0answers
22 views

Two variable delta-finite function

Let ore algebra $\mathbb{O}:=\mathbb{C}(x,y)[D_x , 1,D_x][D_y,1,D_y]$ Let $F(x,y):=\sum_{m,n}a_{m,n}x^m y^n$ is a $\partial$ finite function in two variable over the field of rational function $k:=\...
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227 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) $ \...
4
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1answer
136 views

Are descents in alternating subgroup counted by $h$-vector?

Consider the alternating subgroup $A_n$ of the symmetric group $S_n$ (or in general any Coxeter Group). Is there a simplicial complex whose $h$-vector $h_i$ equals the number of elements of $A_n$ with ...
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94 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 $b_n$, $...
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147 views

Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
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2answers
218 views

Finite lattice representation problem checking

[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem ...
4
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1answer
202 views

Decompostion of hook schur function in terms of cauchy product of holonomic functions

Let $s_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character. \begin{...
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2answers
141 views

Evaluation of irreducible representations of the hyperoctahedral group at bipartition $(\lambda,\mu)=([n],\emptyset)$

There is a very simple formulation for the character of irreducible representations of $S_n$ evaluated on an n-cycle, i.e. that it is 0 on all non-hook partitions, and $(-1)^m$ on hooks. Is there an ...
1
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1answer
83 views

Polarization operators and the action of $GL_{\ell}(\mathbb{R})$ on $\mathcal{R}_{n}^{(\ell)}$

(Also in Mathematics stack Exchange: https://math.stackexchange.com/questions/2528216/polarization-operators-identity-and-gl-ell-mathbbr) Let $X$ be a matrix of variables $x_{ij}$ of size $\ell\...
4
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1answer
110 views

Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?

Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$. The $n$-th type-A subdivision ...
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175 views

Characterizing $n$-exceptions of the ring of symmetric polynomials

(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials) We say that an homogeneous symmetric polynomial ...
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203 views

Computing the equivariant cohomology class of a Białynicki-Birula cell

One of my current research interests is Hessenberg varieties. Briefly, if $m_1\le m_2\le \cdots \le m_{n-1}$ is a weakly increasing sequence of positive integers such that $i\le m_i\le n$ for all $i$, ...
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0answers
67 views

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 ...
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287 views

A spin extension of a Coxeter group?

Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$. Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...
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2answers
1k views

Have you seen my matroid?

Let $M(n,k)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$. Note that the signed permutation group (the ...
0
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1answer
113 views

Asymptotic holonomic

Let $$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}e^{d(k-\frac{d+1}{2})h} . $$ We claim that $ F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}$ is not holonomic??? I want to prove that above thing. Which ...
21
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2answers
844 views

Why is the catalecticant invariant under coordinate changes?

Let $\mathbf{k}$ be a commutative $\mathbb{Q}$-algebra. (We could play the same game over any commutative ring $\mathbf{k}$, but this would be a bit more technical, so let me avoid it.) Fix a ...
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153 views

Non-boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-boolean Eulerian lattice is the following: It ...
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1answer
109 views

Is an Eulerian subgroup lattice boolean?

Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$. The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \...
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1answer
304 views

Existence of a non-Eulerian atomistic lattice with this property on the Möbius function

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$. Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...
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1answer
336 views

Detailed modern references for basic properties of Pfaffians over commutative rings

Pfaffians are important to algebraic combinatorics, at least. This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing ...
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112 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}...
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254 views

A stronger version of a problem of Kenneth Brown using representations

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$. The reduced Euler characteristic of the order complex of the coset poset $\{ ...
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347 views

Internal tensor product of strict polynomial functors: is there a more explicit definition?

In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
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488 views

Categorifying the Cauchy kernel as a filtration of $\operatorname*{Sym}\left( F\otimes G\right) $ over any commutative ring

Question 1 (short version). Let $R$ be a commutative ring with unity. Let $F$ and $G$ be two $R$-modules. Let $n\in\mathbb{N}$. Is it true that the $n$-th symmetric power $\operatorname*{Sym}\...
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0answers
86 views

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-...
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170 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 $\...
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1answer
138 views

Counting cosets in the Quotient of Weyl groups

Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its ...
5
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1answer
214 views

Divisibility labeling on a boolean lattice and nonzero Euler totient

Let $B_n$ be the subset lattice of $\{1,2, \dots , n \}$, also called the boolean lattice of rank $n$. A labeling $f: B_n \to \mathbb{N}_{\ge 1}$ is called acceptable if $\forall a,b \in B_n$: ...
11
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1answer
296 views

Reference request: Heyting algebra structure on Catalan numbers

I've noticed that for every natural number $n\in\mathbb{N}$, there is a finite Heyting algebra with cardinality $C(n)$, where $C(n)$ is the $n$th Catalan number, $$1,1,2,5,14,42,132,\ldots$$ I'm ...
3
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1answer
177 views

Is Leray's theorem on commutative Hopf algebras proven in Milnor-Moore?

Question 1. Is a correct proof of Leray's theorem (the one that says that a connected graded Hopf algebra $H$ over a field of characteristic $0$ is isomorphic as an algebra to the symmetric ...
12
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2answers
558 views

Determinants: periodic entries $0,1,2,3$

Consider an $n\times n$ matrix $M_n$ where the sequence $$\{1,2,3,\dots,n^2\} \mod 4=\{1,2,3,0,1,2,3,\dots\}$$ forms a clock-wise spiral, in that given order. For example, $$M_4=\begin{bmatrix} 1&...