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Let $A_1 \subset A_2 \subset \cdots$ and each $A_i$ is a finite set of polynomials over variables $x_1, \ldots, x_n$. For each $i$, let $N_i$ be the Newton polytope of $A_i$. Since $A_{\infty}$ has in …

$\DeclareMathOperator\Gr{Gr}$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 coordina …

There is a symmetric group action on the power set of positive roots in type A. The action is defined as follows. Denote by $\alpha_1, \ldots, \alpha_n$ be the set of simple roots in a root system. In …

Given two $k$-subsets $I$ and $J$ of $\{1 \dots n\}$, denote by $\min(J)$ the minimal element in $J$ and by $\max(I)$ the maximal element in $I$, we write $I \prec J$ if $\max(I)<\min(J)$. The sets $I …

Rigid indecomposable modules in the category ${\rm CM}(A)$ of Cohen-Macaulay $A$-module are parametrized by certain arrays of integers called profiles as shown in the paper A categorification of Grass …

Given $v_{ij} \in \{0,1\}$, $i \in \{1,2\}$, $j \in \{1,2,\ldots,n\}$. Let $X_1, X_2, \ldots, X_n$ be random variables, $P[X_i=1]=P[X_i=0]=1/2$, $i \in \{1,\ldots, n\}$. By checking many examples, I t …

Let $k \le n$ be positive integers and let $m$ be a positive integer. Assume that $x_1, \ldots, x_n$ are non-negative integers and \begin{align} & x_1^2 + x_2^2 + \cdots + x_n^2 - (k-2) m^2=2, \\ & x_ …

Denote by $SSYT(\lambda, [n])$ the set of all semi-standard Young tableaux of shape lambda with entries in $[n]=\{1, \ldots, n\}$. Denote by $SSYT(\lambda, \infty)$ the set of all semi-standard Young …

In my research, I need to find the set of all minimal dependent sets of a given set of vectors. One method is to check every subset of the given set. But this method is very slow when the set of vecto …

Is there some software which computes Hilbert series of quotients of exterior algebras? In commutative case, Maple can compute Hilbert series. Thank you very much.

Let $\Phi_+$ be the set of all positive roots for a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root as …

Let $\mathbb{Z}_m = \mathbb{Z}/m\mathbb{Z}$. Let $A$ be an $k \times n$ matrix over $\mathbb{Z}_m$. Let $f: \mathbb{Z}_m^n \to \mathbb{Z}_m^k$ be a linear map defined by $f(x) = Ax$, $x \in \mathbb{Z} …

Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker rel …

Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. Then $Gr(k,n)$ is a projective variety and it has Plucker coordinates $P_{i_1, \ldots, i_k}$ ($i_1<\ldots< …

Are there some generalized Catalan number of type $B$ such that the sequence of the numbers is $3,9,29,97,333$ for $n=2,3,4,5,6$? As discussed in this previous question, there are at least two types …