I need either a proof or a reference in modern (scheme-theoretic) language. According to Sansuc, this result can be gleaned from Borel's book on linear algebraic groups, but the old-style algebraic geometry language makes my head spin. Isn't there a modern proof of the above statement somewhere, for God's sake?

Let $V_m = span\{v_1, \ldots, v_m\}$ be the $m$-dimensional standard $U_q(gl_n)$-module. How to write down the actions $E_i.v_j=$, $F_i.v_j=$, and $K_i.v_j=$?

In $gl_n$ case, we have $v_i = (0, \ldots, 0, 1, 0, \ldots, 0)^T$, $1$ is in the $i$-th place. The actions are given by $E_i.v_j=\delta_{ij} v_i$.

I don't know the case of $U_q(gl_n)$. Are there some references which describe the action? Thank you very much.

Are there some references which write presentations of quantum general linear groups explicitly? For example, what are generators and relations for $U_q(gl_2)$? Thank you very much.

I'm reading Gelbart's **Introduction to the Selberg Trace Formula** https://arxiv.org/abs/math/0407288. In his paper he seems to have used the consequence that a smooth function with compact support is a Schwartz-Bruhat function. I'm wondering whether this is correct.

Let $T$ be a closed rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of $G$ onto the unit disk $\mathbb{D}.$

My question is the following: for $n\in \mathbb{N},$ $a\in \mathbb{D},$ and $w\in G\setminus \{\Phi^{-1}(a)\},$ find the residue of $$ \frac{1}{(\Phi(z)-a)^n(\Phi'(z))^{1/2}(z-w)}$$ at $\Phi^{-1}(a).$

Definitely, we can apply the limit formula for higher order poles: $$\frac{1}{(n-1)!}\lim_{z\rightarrow \Phi^{-1}(a)} \frac{d^{n-1}}{dz^{n-1}}\frac{(z-\Phi^{-1}(a))^n}{(\Phi(z)-a)^n(\Phi'(z))^{1/2}(z-w)}.$$ However, finding the above limit is quite messy (I use, for example, Leibniz’s formula). I would like to know if there's any technique in complex analysis or conformal mapping theory that can help to solve this question easily.

Thank you very much.

How to prove that

in $A_n$ (Alternating group), the subgroup of second smallest index has index $n \choose 2$ if $n\ge 9$ ?

I know how to prove it for the smallest index, but for the second smallest I don't know how to prove it.

A braided vector space is a pair $(V, \Psi)$, where $V$ is a vector space and $\Psi$ is an invertible linear operator on $V \otimes V$ such that $\Psi_1 \Psi_2 \Psi_1 = \Psi_2 \Psi_1 \Psi_2$. The map $\Psi$ is called a braiding. Let $(V, \Psi)$ be a braiding vector space and $V=V_1 \oplus V_2$. Under what condition will we have $\Psi_{V_1 \oplus V_2, V_1 \oplus V_2} \circ \Psi_{V_1 \oplus V_2, V_1 \oplus V_2} = \oplus_{i,j=1}^2 (\Psi_{V_i, V_j} \circ \Psi_{V_j, V_i})$? Here $\Psi_{V_i, V_j} = \Psi_{V,V}|_{V_i \otimes V_j}$. Thank you very much.

First, I'm not sure this question is suitable for MO. As a quick Google search gave no relevant result, I decided to ask it here.

Let $ F : s\mapsto\sum_{n>0}\dfrac{f_{n}}{n^{s}} $ whenever $ \Re(s)>1 $ be the Dirichlet series of an L-function, and let's define the 'tensor product' $ F\otimes G $ of $ F $ and $ G : =s\mapsto\sum_{n>0}\dfrac{g_{n}}{n^s} $ for $ \Re(s)>1 $ defined as follows :

$ F\otimes G : s\mapsto\sum_{n>0}\dfrac{f_{n}g_{n}}{n^{s}} $ for $\Re(s)>1 $ .

Let's also define for any L-function $ F $ the $ \alpha $-deformation of $ F $ for any non negative real number $ \alpha $ less or equal to $ 1 $, denoted by $ F_{\alpha} $, as $F_{\alpha} : s\mapsto\sum_{n>1}\dfrac{f_{n}^\alpha}{n^s} $ for $ s $ in some right half plane where convergence issues don't matter.

Consider now a map $\phi $ from the set of all L-functions to itself such that for a given L-function $ F $, two L-functions $ G $ and $ H $ exist such that $ G=\phi(F) $ and $ H=\phi(G) $ .

Let's define $ \phi^{(\alpha)}(F) $ by $ \phi^{(\alpha)} (F): s\mapsto\sum_{n>0}\dfrac{f_{n}^{1-\alpha}g_{n}^{\alpha}}{n^{s}} $ in a suitable right half plane and $ \phi^{(\alpha)}(G) $ similarly.

Can we prove rigorously that $ \phi^{(\alpha)}(F\otimes G)=F_{1-\alpha}\otimes G\otimes H_{\alpha} $ ? If so, would such a result be useful in analytic number theory ? Have such ideas been considered so far ?

Let $\Omega$ be a $C^1$ domain (or a domain that has the necessary regularity) and consider the function space $S_{\epsilon,p}:=\{ u \in W^{1,p(1-\epsilon)}_0(\Omega) : \|\nabla u\|_{L^{{p}(1-\epsilon)}(\Omega)} =1\}$ for some $0 < \epsilon < 1$ and any $p$ such that $p(1-\epsilon)>1$. Define the following quantity $$ m_{p,\epsilon}(u) = \inf_{\phi \in W_0^{1,p}(\Omega)}\| |\nabla u|^{-\epsilon} \nabla u - \nabla \phi \|_{L^p(\Omega)}$$

It is easy to see that by taking $\phi \equiv 0$, that $m_{p,\epsilon}(u)\leq 1$ for any $u \in S_{p,\epsilon}$ and $\epsilon \in (0,1)$.

Show that $m_{p,\epsilon}(u) \leq \delta(\epsilon,p) < 1$ for any $u \in S_{p,\epsilon}$.

I can show this for any $p>1$ and $\epsilon \in (0,\epsilon_0)$ for some very small $\epsilon_0$. How can I show this for all $\epsilon \in (0,1)$?

By convexity of the functional, it is easy to see that $\phi$ in the definition which minimizes the norm is unique.

Suppose $\cal{C}$ is a small stable $\infty$-category. Then, we have its K-theory spectrum $K(\cal{C})$ that gives us K-theory groups $K_n(\cal{C})$ by taking stable homotopy groups. There are criteria relating the vanishing of $K_{-1}(\cal{C})$ to the existence of t-structures on $\cal{C}$. Are there "easy-to-use" criteria for the vanishing of $K_1(\cal{C})$?

Let $W(t)$ be a standard brownian motion in $E \triangleq \mathbb{R}^d$. The transition probability from a state $x \in E$ at time $t$ to a state $y \in E$ at time $T$ is $$ p(x,t;y,T) = \frac{1}{\sqrt{2 \pi}^d} \exp \left ( -\frac{1}{2} \frac{\|x-y \|^2}{T-t}\right ). $$ Using the Chapman-Kolmogorov equation, it is clear that for all $n \geq 0$ and for all finite sequence of states $x_0 = x, x_1, ... , x_n, x_{n+1} = y$ and $t_0 = t < t_1 < ... < t_n < t_{n+1} = T$, we have $$ p(x,t;y,T) = \int_{E^n} \frac{1}{\sqrt{2 \pi}^{nd}} \exp \left ( -\frac{1}{2} \sum_{i=0}^{n} \frac{\|x_{i+1}-x_i \|^2}{t_{i+1}-t_i}\right ) d x_1 ... d x_n. $$ I am not so sure that the limit as $n \to \infty$ makes sense in the right hand side above but I believe that it can be interpreted in terms of the Wiener measure on $\mathcal{C}([0,\infty);E)$ and $$ C(x,t;y,T) \triangleq \{ \omega \in \mathcal{C}([0,\infty);E) \: | \: w(t) = x, \: w(T) = y \}. $$ (if no mistake) how to interpret $p(x,t;y,T)$ in terms of $\mu$ and $C(x,t;y,T))$?

Let $G$ be a finite cyclic group and $\widehat{G}$ the character group. Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ for any $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $H_{S} \subset \widehat{G}$ be the subgroup generated by $S$.

We now consider a sequence $(G_{i},S_{i})$ as above with $|G_{i}|\rightarrow \infty$. Suppose that there exists $\epsilon$ with $ 0 < \epsilon < 1$ such that $|H_{S_{i}}| \gg |G_{i}|^{1-\epsilon}$. Then, is it necessary to have $|S_{i}| \gg |G_{i}|^{\epsilon}$? Can we say something about optimal lower bound?

I have a question about Poincaré inequalities.

It is known that Poincaré inequalities is related to heat kernel estimates on Euclidean domains, manifolds, metric measure spaces. (I am especially interested in Euclidean case, though). Let $\Omega$ be a domain on $\mathbb{R}^{d}$. I am interested in the following two types of inequality: there exists two positive constants $C_{1}, C_{2}$, which may depend on $\Omega,d$, such that \begin{align*} & \int_{\Omega}|f-f_{\Omega}|^{2}\,dx\le C_{1} \int_{\Omega}|\nabla f|^{2}\,dx\cdots(1),\\ &\int_{\tilde{B}(x,R) }|f-f_{\tilde{B}(x,R)}|^{2}\,dx\le C_{2} \int_{\tilde{B}(x,R)}|\nabla f|^{2}\,dx\cdots(2), \end{align*} where $\tilde{B}(x,R)=B(x,R) \cap \Omega$, $x \in D,R>0$, $f \in H^{1}(\Omega)$ and \begin{equation*} f_{E}:=\frac{1}{m(E)} \int_{E}f\,dx\text{, $m$ is the Lebesgue measure}. \end{equation*}

**Question:**

Can we comfirm $(1) \Rightarrow(2)$? In this note(enter link description here), to obtain a nice heat kernel estimate, condition (2) is imposed. By the way, we can comfirm the following inequality:
\begin{equation*}
\int_{\tilde{B}(x,R) }|f-f_{\tilde{B}(x,R)}|^{2}\,dx\le C_{3} \int_{\tilde{B}(x,R)}|\nabla f|^{2}\,dx\cdots(3).
\end{equation*}
But $C_{3}$ **may depend on** $x,R$.

I am reading research papers related to Graph Isomorphism, while reading those things I encountered two theorems given below.

How to prove that

**Theorem 1** : Let $G \le S_n$ be a primitive group other than $S_n$ or $A_n$. If $n$ is sufficiently large then $|G| < n^{\sqrt{n}}$

**Theorem 2:** Let $G \le S_n$ be a primitive group other than $S_n$ or $A_n$. If $n$ is sufficiently large then $|G| < $ exp$(4{\sqrt{n}} \ln^2 n)$

This is a consequence of the classification of finite simple groups (*Peter J Cameron*). I am not able to prove theorem 1 and 2 given above.

Thank You.