Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lam …

They had plenty of time to adopt the theory of categories. They had Eilenberg, then Cartan, then Grothendieck. Did they feel that they have established their approach already, that …

Here is the text of Exercise: 2 a) Let $X$ be an ordered set. Show that the set of intervals $\left[x, \rightarrow\right[$ (resp. $\left]\leftarrow, x\right]$) is a base o …

Let $\bf C$ be a category with pullbacks. Define $Span\colon (A,B)\mapsto \{ (X,f,g)\mid A\xleftarrow{f}X\xrightarrow{g}B\}$ and notice that it is a profunctor $s\colon \bf C^\text …

In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's Catalan exercises in EC2 a …

Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$. Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes. My q …

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$? I am not necessarily looking for an explicit description of the canonical mod …

Let $X$ be a connected, integral curve over a field $k$, and let $Y \rightarrow X$ be a finite etale cover. Corresponding to this cover there is a finite extension of function fiel …

I have to solve exercise 5.4 and 5.6 of the book Bayesian computation with R. I don't understand the iter to solve this kind of exercise. For instance how do I reparametrize $\be …

For a given partition $[n_{1},...,n_{k}]$ of $N \in \mathbb{N}$ there exists a corresponding nilpotent orbit variety $O_{[n_{1},...,n_{k}]}$ in $\mathfrak{gl}(N)$ which can be repr …

Let $$J_1=\int_0^1\frac{1}{\sqrt{1-t_2}}dt_2,$$ $$J_2=\int_0^1 \int_0^{t_2}\frac{1}{\sqrt{1-t_2}}(\frac{1}{\sqrt{1-t_3}}+\frac{1}{\sqrt{t_2-t_3}})dt_3dt_2,$$ $J_3=\int_0^1 \int_0^{ …

Let $X$ be a normal complex projective variety, let $V$ be a closed subset of $X$ (possibly reducible), and let $I_V$ be its ideal sheaf (consider the reduced scheme structure for …

Let $s \in R^{n}$ (meaning $s$ is $n \times 1$ vector), where $n$ is the dimension of the vector. The ideal sliding term, $\nu$ is taken to be: \begin{equation} \nu = \frac …

Let $X$ be a general chain of $d$ lines in $\mathbb P^n$, where $n \geq 3$. Let $I$ be the homogeneous ideal of polynomials vanishing on $X$. What is the Hilbert function $$P(k) = …

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows …