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Let $R$ be a ring and $R((x))$ the ring of formal Laurent series. The elements in the ring $R((x))$ are series of the form $$ f = \sum_{n\in\mathbb{Z}} a_n x^n, $$ where ${\displaystyle a_{n}=0}$ for …

Let $f= xy^3+y^4-x^2+xy$. Using the following codes in Maple, f := xy^3+y^4-x^2+xy; v := Weierstrassform(f, x, y, x0, y0); I obtain the following result: \begin{align} & f_0 = {{ x_0}}^{3}+{{ y_ …

Let $A$ be a finite dimensional algebra generated by $x_1, \ldots, x_n$ subject to certain relations $I_1, \ldots, I_m$. Could we obtained a linear basis $B$ consisting of monomials in $x_1, \ldots, x …

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} P_{i …

The exchange relation in a cluster algebra is \begin{align} x_k' = \frac{1}{x_k} (\prod_{j \to k}x_j + \prod_{k \to j} x_j). \end{align} Do we have some tropical version of this relation? Are there so …

In the paper, the formula (2.4) gives a tropical version of mutation relations: \begin{align} a_k' = \max( \sum_i a_i[b_{ki}]_+, \sum_i a_i [-b_{ki}]_+ )-a_k. \end{align}

Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ and $I$ a non-homogeneous ideal of $R$. Then the algebra R/I is filtered. It has an associated graded algebra gr(R/I). Let $I_1$ be the initial ideal of $I$, …

In THE SLOPES DETERMINED BY n POINTS IN THE PLANE by JEREMY L. MARTIN, Page 2, Theorem 1.1, a Hilbert series is used to compute some combinatorial objects: Let $R_n=k[m_{1,2}, \ldots, m_{n-1,n}]$, a …

Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates) \begin{align} & {p_1}^2 {p_2}^2 c_1 {t_1}^2 t_2 + …

Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$: $$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and …

In the paper, cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have cl …

Super Grassmannians are introduced by Manin, see for example. We have Plucker relation for Grassmannian. Are there some references about super Plucker relations for super Grassmannian? Thank you ver …

$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $G=\SL_k$ be the special linear group, $U$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $U^+$ the unipotent subgroup …

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat decompo …

I am reading the lecture notes INTRODUCTION TO DONALDSON–THOMAS INVARIANTS. I have a question in the end of page 1 about the proof of a map is an automorphism. Let $m>0$ be an integer. Let $\overline …