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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 …

Catalan numbers are generalized to type B: https://oeis.org/A000984. Are there some references about Catalan numbers of type C, D? 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 …

In the paper, geometric RSK correspondence is given by $$ \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & \frac{ad}{b+c} \end{matri …

In the lecture notes, it is said that (Theorem 3.1.3) the set of positroid cells in $Gr(k,n)$ are in one to one correspondence with the set of bounded affine permutations of type $(k,n)$. In Example 4 …

Let $v_1,v_2 \in \{0,1\}^n$. Denote $v_1v_2=((v_1)_1 (v_2)_1, \ldots, (v_1)_n (v_2)_n)$ and $|v|=\sum v_{i}$. \begin{align} {\scriptsize f(v_1, v_2) = \sum_{x_1=0}^{|v_1|} \sum_{x_2=0}^{|v_2|} \sum_{d …

I am trying to understand the proof of Laurent phenomenon of cluster algebras in the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras. Chapters 1-3, arXiv:1608. …

In the paper The Amplituhedron , Nima Arkani-Hamed and Jaroslav Trnka introduced the geometric object amplituhedron. It is defined as follows (see also the lecture notes). Let $Z$ be a $(k+m)\times …

Let $S_n$ be the symmetric group on $\{1, \ldots, n\}$. Let \begin{align} T=\sum_{g\in S_n} g. \end{align} Are there some references about the factorization of $T$? In the case of $n=3$, we have \b …

Are there some references which proves the following result? Let $W$ be a Coxeter group and $w \in W$. Then different reduced expressions of $w$ can be transformed from one into anther using only the …

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 …

For $v, w \in \{0,1\}^n$, denote $v w = (v_1 w_1, \ldots, v_n w_n)$ and $|v|=\sum_{i} v_i$. Let $v_1, v_2 \in \{0,1\}^n$ and \begin{align*} f(x_1, x_2) = \sum_{d=0}^{|v_1 v_2|} \frac{1}{2^{|v_1|+|v_ …

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 …

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_ …

In the webpage, there is a result: Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $F_ …